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BASIC STATISTICS FOR MANAGEMENT
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BASIC STATISTICS FOR MANAGEMENT
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UNIT 1
INTRODUCTION TO STATISTICS
Statistics and Rationale, Frequency Distribution, Classification and Tabulation,
Diagrammatical and Graphical Representation.
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MEANING
The word Statistics describes several concepts of importance to decision -maker. It is
important for a beginner to have an understanding of these different concepts.
STATISTICAL METHODS V/S EXPERIMENTAL METHODS
We try to get the knowledge of any phenomenon through direct experiment. There may be
many factors affecting a certain phenomenon simultaneously. If we want to study the effect
of a particular factor, we keep other factors fixed and study the effect of only one factor. This
is possible in all exact s ciences like Physics, Chemistry etc. This method cannot be used in
many sciences, where all the factors cannot be isolated and controlled. This difficulty is
particularly encountered in social sciences, where we deal with human beings. No two
persons are e xactly alike. Besides the environment also changes and it has its effect on every
human being and therefore it is not possible to study one factor keeping other conditions
fixed. Here we use statistical methods. The results obtained by the use of this scie nce will not
be as accurate as those obtained by experimental methods. Even then they are of much use
and they have a very important role to play in the modern World. Even in exact sciences
some of the statistical methods are made use of.
The word Statisti cs is derived from the Latin word "statis' which means a political state. The
word Statistics was originally applied to only such facts and figures that were required by the
state for official purposes. The earliest form of statistical data is related to c ensus of
population and property, through the collection of data for other purposes was not completely
ruled out. The word has now acquired a wider meaning.
STATISTICS IN PLURAL
Statistics in plural refer to any set of data or information. The president of a company may
call for 'statistics on the sales of northern region' or an MP may quote the statistics on price -
rise in agricultural products. More familiar examples for the students will be the marks of
students in a class, the ages of children in primary school.
Prof. Secrist defines the word 'Statistics' in the first sense as follows"
"By Statistics we mean aggregate of facts affected to a marked extend by multiplicity of
causes, numerically expressed, enumerated or estimated according to reasonable stan dards of
accuracy, collected in a systematic manner for a predetermined purpose and placed in relation
to each other."
This definition gives all the characteristics of Statistics:
i. Aggregate of Facts: A single isolated figure is not 'Statistics.' Marks o f one student in one
subject will not called Statistics. But, if we consider the marks of all the students in the class
in a particular subject, they will be called 'Statistics.'
ii. Affected by Multiplicity of causes: There are various causes for the chan ges in the data,
the marks of the students depend upon, the intelligence of students, their capacity and desire
to work etc.
iii. Numerically expressed: Unless the characteristics have some numerical measurement
they will not be called Statistics. The stat ement 'A student writes very good English' is not
Statistics. But if marks of the whole class in 'English' are given they will be called 'Statistics.' munotes.in
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iv. Enumerated or Estimated according to reasonable standards of accuracy: However
much a person tries, i t is not possible to attain perfect accuracy whether we actually measure
or estimate the characteristic. But a certain standard of accuracy should be set up according to
the problem under consideration. The estimate for the cost of big project may be corre ct up to
Rs. 1, 000 but for household expenses it should be correct up to a rupee.
v. Collected in a systematic manner: There should be a method in the manner of collection,
then only the figures will be reliable and useful.
vi. Collected for a predetermin ed purpose: Unless we know the purpose, the data collected
may not be sufficient. Besides some unnecessary information may be collected which will be
a waste of time and money.
vii. Placed in relation to each other: Only when we want to compare characteris tics, which
have some relation with each other, we collect Statistics. The wages of fathers and ages of
sons should not be collected together. But we can have ages and heights of a group of
persons, so that we can find the relation between the two.
STATIS TICAL METHODS
The word Statistics used in the second sense means the set of techniques and principles for
dealing with data.
1. Suppose you have the data about production profits and sales for a number of years of a
company. Statistics in this sense is concerned with questions such as
(i) What is the best way to present these data for review?
(ii) What processing is required to reveal more details about the data?
(iii) What ratios should be obtained and reported?
2. A public agency wants to estimate the number of fish in a lake. Five hundred fish are
captured in a net tagged and returned to the lake. One week later 1, 000 fish a re captured
from the same lake in nets and 40 are found to be with tags. Here Statistics in this second
sense deals with questions such as:
(i) What is a good estimate of the number of fish in the lake?
(ii) What is our confidence in it and how much error can be expected?
and (iii) Can we have a method, which will make a better estimate?
Statisticians have defined this in various ways. Bowley says, "Statistics may rightly be called
the science of averages." But this definition is not correct. Statistics has many refined
techniques and it does much more than just averaging the data.
Kendall defines it as, "The branch of scientific methods that deals with the data obtained by
counting or measuring the properties of population of natural phenomena." This defini tion
does not give the idea about the functions of Statistics. It is rather vague.
Seligman defines it as, "The science which deals with the methods of collecting, classifying,
presenting, comparing and interpreting numerical data collected to throw some l ight on any
sphere of inquiry." Croxton , Cowden and Klein define it as, "The last two definitions can be
considered to be proper which explain the utility of 'statistics'. We will examine the four
procedures mentioned in the definition in brief.
Collection : The day may be collected from various published and unpublished sources, or the
investigator can collect his own information. Collecting first hand information is a very
difficult task. The usefulness of the data collected depends to a very great extent upon the munotes.in
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manner in which they are collected. Though theoretical knowledge is necessary for the proper
collection o f data, much can be learnt through experience and observation.
Presentation: The data collected, to be understood, should be presented in a suitable form.
Just a given mass of figures signifies nothing to a person and they can lead only to confusion.
They are usually presented in a tabular form and represented by diagrams.
Analysis: Presentation of data in a tabular form is one elementary step in the analysis of the
collected data. If we want to compare two series, a typical value for each series is to be
calculated. If we want to study some characteristic of a big group, exhaustive study is not
possible. We take a sample, study it and inferences are drawn on the basis of sample studies.
Sometimes forecasting is necessary. The management of a firm may be int erested in future
sales. For that it has to analyse the past data. We are going to study some of these methods of
analysing the data in this book.
Interpretation: This is the final step in an investigation. Based upon the analysis of the data,
we draw cert ain conclusions. While drawing these conclusions, we must consider that nature
of the original data. Experts in the particular field of activity must make the final
interpretation. The statistical methods are not like experimental methods, which are exact.
For interpreting the analysis of the data dealing with some psychological problems, a
psychologist is right person. (An economist, though well versed in statistical methods will not
be of any use there).
STATISTICAL MEASURES
Statistics also has a precise technical meaning. Measures derived from the sample data are
referred to as Statistics. If only one measure is obtained it is called a Statistic.
A magazine takes a sample of 100 readers. 15 of them are over 30 years of age. The sample
proportion of reade rs over 30 years of age is 0.15. This sample proportion is referred to as a
statistic obtained by this survey.
The weekly sales for 5 weeks for a salesman are Rs. 2, 000, Rs. 2, 500, Rs. 15, 000, Rs. 3000
and Rs. 1, 800. As a measure of the spread of the v alues the difference between the smallest
and the largest value (called the range) is calculated. This range is a statistic.
IMPORTANCE OF STATISTICS
Statistics is not studied for its own sake. It is employed as a tool to study the problems in
various nat ural and social sciences. The analysis of data is used ultimately for forecasting,
controlling and exploring.
Statistics is important because it makes the data comprehensible. Without its use the
information collected will hardly be useful. To understand t he economic condition of any
country we must have different economic aspects quantitatively expressed and properly
presented. If we want to compare any two countries, statistics is to be used. For studying
relationship between two phenomena, we have to tak e the help of statistics, which explains
the correlation between the two.
People in business can study past data and forecast the condition of their business, so that
they can be ready to handle the situations in future. Nowadays a businessman has to deal with
thousands of employees under him and cannot have direct control over them. Therefore, he
can judge them all and control their performance using statistical methods e.g., he can set up munotes.in
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certain standards and see whether the final product conforms to the m. He can find out the
average production per worker and see whether any one is giving less, i.e., he is not working
properly.
Business must be planned properly and the planning to be fruitful must be based on the right
analysis of complex statistical data . A broker has to study the pattern in the demand for
money by his clients, so that he will have correct amount of reserves ready.
Scientific research also uses statistical methods. While exploring new theories, the validity of
the theory is to be tested o nly by using statistical methods. Even in business many new
methods are introduced. Whether they are really an improvement over the previous ones, can
be tested using statistical techniques.
We can see many more examples from almost all sciences, like biol ogy, physics, economics,
psychology and show that statistical methods are used in all sciences. The point here is that
'Statistics' is not an abstract subject. It is a practical science and it is very important in the
modern World.
FUNCTIONS OF STATISTICS
1. Statistics presents the data in numerical form: Numbers give the exact idea about any
phenomenon. We know that India is overpopulated. But only when we see the census figure,
548 millions, we have the real idea about the population problem. If we want to compare the
speed of two workmen working in the same factory, with the same type of machine, we have
to see the number of units they turn out every day. Only when we express the facts with the
help of numbers, they are convincing.
2. It simplifies the c omplex data: The data collected are complex in nature. Just by looking
at the figures no person can know the real nature of the problem under consideration.
Statistical methods make the data easy to understand. When we have data about the students
making u se of the college library, we can divide the students according to the number of
hours spent in the library. We can also see how many are studying and how many are sitting
there for general reading.
3. It facilitates comparison: We can compare the wage con ditions in two factories by
comparing the average wages in the two factories. We can compare the increase in wages and
corresponding increase in price level during that period. Such comparisons are very useful in
many social sciences.
4. It studies relatio nship between two factors: The relationship between two factors, like,
height and weight, food habits and health, smoking and occurrence of cancer can be studied
using statistical techniques. We can estimate one factor given the other when there is some
relationship established between two factors.
5. It is useful for forecasting: We are interested in forecasting using the past data. A
shopkeeper may forecast the demand for the goods and store them when they are easily
available at a reasonable price. He ca n store only the required amount and there will not be
any problem of goods being wasted. A baker estimates the daily demand for bread, and bakes
only that amount so that there will be no problem of leftovers.
6. It helps the formulation of policies: By st udying the effect of policies employed so far by
analysing them, using statistical methods, the future policies can be formulated. The munotes.in
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requirements can be studied and policies can be determined accordingly. The import policy
for food can be determined by s tudying the population figures, their food habits etc.
LIMITATIONS OF STATISTICS
Though Statistics is a very useful tool for the study of almost all types of data it has certain
limitations.
1. It studies only quantitative data: A very serious drawback is that statistics cannot study
qualitative data. Only when we have data expressed in numerical form we can apply
statistical methods for analysing them. Characteristics like beauty, cruelty, honesty or
intelligence cannot be studied with the help of statist ics. But in some cases we can relate the
characteristics to number and try to study them. Intelligence of students can be studied by the
marks obtained by them in various tests, we can compare the intelligence of students or
arrange them in order if we tak e marks as an indicator of intelligence. Culture of a society or
the lack of it can be studied considering the number of charitable institutions, their sizes and
number of crimes.
2. It cannot be used for an individual: The conclusions drawn from statistical data are true
for a group of persons. They do not give us any knowledge about an individual. Though
Statistics can estimate the number of machines in a certain factory that will fail after say, 5
years, it cannot tel l exactly which machines will fail. One in 2, 000 patients may die in a
particular operation. Statistically this proportion is very small and insignificant. But for the
person who dies and his family, the loss is total. Statistics shows now sympathy for su ch a
loss.
3. It gives results only on an average: Statistical methods are not exact. The results obtained
are true only on an average in the long run. When we say that the average student studies for
2 hours daily there may not be a singly student studyin g for 2 hours, not only that, every day
the average will not be 2 hours. In the long run, if we consider a number of students, the daily
average will be 2 hours.
4. The results can be biased: The data collected may sometimes be biased which will make
the w hole investigation useless. Even while applying statistical methods the investigator has
to be objective. His personal bias may unconsciously make him draw conclusions favourable
in one way or the other.
5. Statistics can be misused: It is said that statis tics can prove or disprove anything. It
depends upon how the data are presented. The workers in a factory may accuse the
management of not providing proper working conditions, by quoting the number of accidents.
But the fact may be that most of the staff i s inexperienced and therefore meet with an
accident. Besides only the number of accidents does not tell us anything. Many of them may
be minor accidents. With the help of the same data the management can prove that the
working conditions are very good. It can compare the conditions with working conditions in
other factories, which may be worse. People using statistics have to be very careful to see that
it is not misused.
Thus, it can be seen that Statistics is a very important tool. But its usefulness depe nds to a
great extent upon the user. If used properly, by an efficient and unbiased statistician, it will
prove to be a wonderful tool.
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BRANCHES IN STATISTICS
Statistics may be divided into two main branches:
1. Descriptive Statistics: In descriptive stat istics, it deals with collection of data, its
presentation in various forms, such as tables, graphs and diagrams and findings, averages and
other measures which would describe the data.
For example, Industrial Statistics, population statistics, trade stati stics etc....Such as
businessmen make to use descriptive statistics in presenting their annual reports, final
accounts and bank statements.
2. Inferential Statistics: In inferential statistics deals with techniques used for analysis of
data, making the est imates and drawing conclusions from limited information taken on
sample basis and testing the reliability of the estimates.
For example, suppose we want to have an idea about the percentage of illiterates in our
country. We take a sample from the populatio n and find the proportion of illiterates in the
sample. This sample proportion with the help of probability enables us to make some
inferences about the population proportion. This study belongs to inferential statistics.
CHARACTERISTICS OF STATISTICS
1. Statistics are aggregates of facts.
2. Statistics are numerically expressed.
3. Statistics are affected to a marked extent by multiplicity of causes.
4. Statistics are enumerated or estimated according to a reasonable standard of accuracy.
5. Statistics are collected for a predetermined purpose.
6. Statistics are collected in a systematic manner.
7. Statistics must be comparable to each other.
SOME BASIC DEFINITIONS IN STATISTICS
Constant: A quantity which can be assuming only one value is called a constant. It is usually
denoted by the first letters of alphabets a, b, c .
For example value of π = 22/7 = 3.14159.... and value of e = 2.71828....
Variable: A quantity which can vary from one in dividual or object to and other is called a
variable. It is usually denoted by the last letters of alphabets x, y, z.
For example, heights and weights of students, income, temperature, number of children in a
family etc.
Continuous variable: A variable whi ch can assume each and every value within a given
range is called a continuous variable. It can occur in decimals.
For example, heights and weights of students, speed of a bus, the age of a shopkeeper, the life
time of a T.V. etc.
Continuous Data: Data whi ch can be described by a continuous variable is called continuous
data.
For example: Weights of 50 students in a class.
Discrete Variable: A variable which can assume only some specific values within a given
range is called discrete variable. It cannot occ ur in decimals. It can occur in whole numbers.
For example: Number of students in a class, number of flowers on the tree, number of houses
in a street, number of chairs in a room etc...
Discrete Data: Data which can be described by a discrete variable is c alled discrete data. munotes.in
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For example, Number of students in a College.
Quantitative Variable: A characteristic which varies only in magnitude from an individual
to another is called quantitative variable. It can be measurable.
For example, wages, prices, heigh ts, weights etc.
Qualitative Variable: A characteristic which varies only in quality from one individual to
another is called qualitative variable. It cannot be measured.
For example, beauty, marital status, rich, poor, smell etc.
EXERCISE
1. Explain the meaning of statistics.
2. Give a definition of statistics and discuss it.
3. Explain the functions of statistics.
4. What are the limitations of statistics?
5. Define the term Statistics and discuss its characteristics.
6. Enumerate with example some terms of Statistics.
7. Discuss on the different branches of Statistics.
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CHAPTER 1
UNIT 2
DATA: COLLECTION AND PRESENTATION
STATISTICAL DATA
A sequence of observation made on a set of objects included in the sample drawn from
population is known as statistical data.
1. Ungrouped Data: Data which have been arranged in a systematic order are called raw
data or ungrouped data.
2. Grouped Data: Data presented in the form of frequency distribution is called grouped
data.
COLLECTION OF DATA
The first step in any enquiry (investigation) is c ollection of data. The data may be collected
for the whole population or for a sample only. It is mostly collected on sample basis.
Collection of data is very difficult job. The enumerator or investigator is the well trained
person who collects the statist ical data. The respondents (information) are the persons whom
the information is collected.
TYPES OF DATA
There are two types (sources) for the collection of data:
1. Primary Data: The primary data are the first hand information collected, compiled and
published by organisation for some purpose. They are most original data in character and
have not undergone any sort of statistical treatment .
For example, Population census reports are primary data because these are collected,
compiled and published by the population census organisation.
2. Secondary Data: The secondary data are second hand information which are already
collected by someone (organisation) for some purpose and are available for the present study.
The secondary data are not pure in character a nd have undergone some treatment at least
once.
For example, Economics survey of England is secondary data because these are collected by
more than one organisation like Bureau of Statistics, Board of Revenue, the Banks etc.
METHODS OF COLLECTING PRIMARY DATA
Primary data are collected by the following methods:
1. Personal Investigation: The researcher conducts the survey him/herself and collects data
from it. The data collected in this way is usually accurate and reliable. This method of
collecting data i s only applicable in case of small research projects.
2. Through Investigation: Trained investigators are employed to collect the data. These
investigators contact the individuals and fill in questionnaire after asking the required
information. Most of the organisations implied this method. munotes.in
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3. Collection through questionnaire: The researchers get the data from local representation
or agents that are based upon their own experience. This method is quick but gives only
rough estimate.
4. Through Telephone: The researchers get information through telephone. This method is
quick.
METHODS OF COLLECTING SECONDARY DATA
The secondary data are collected by the following sources:
Official: The publications of Statistical Division, Ministry of Finance, the Federal
Bureaus of Statistics, Ministries of Food, Agriculture, Industry, Labour etc....
Semi -Official: State Bank, Railway Board, Central Cotton Committee, Boards of
Economic Enquiry etc....
Publication of Trade Associations, Chambers of Commerce etc....
Technical and Trade Journals and Newspapers.
Research Organisations such as Universities and other Institutions.
DIFFERENCE BETWEEN PRIMARY AND SECONDARY DATA
The difference between primary and secondary data is only a change of hand. The primary
data are the firs t hand information which is directly collected from one source. They are most
original data in character and have not undergone any sort of statistical treatment while the
secondary data are obtained from some other sources or agencies. They are not pure i n
character and have undergone some treatment at least once.
For example, suppose we are interested to find the average age of MS students. We collect
the age's data by two methods; either by directly collecting from each student himself
personally or get ting their ages from the University record. The data collected by the direct
personal investigator is called primary data and the data obtained from the University record
is called Secondary data.
EDITING OF DATA
After collecting the data either from prim ary or secondary source, the next step is its editing.
Editing means the examination of collected data to discover any error before presenting it. It
has to be decided before hand what degree of accuracy is wanted and what extent of errors
can be tolerated in the inquiry. The editing of secondary data is simpler than that of primary
data.
CLASSIFICATION OF DATA
The process of arranging data into homogenous group or classes according to some common
characteristics present in the data is called classification.
For example, the process of sorting letters in a post office, the letters are classified according
to the cities and further arranged according to streets.
BASES OF CLASSIFICATION
There are four important bases of classification: munotes.in
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1. Qualit ative Base
2. Quantitative Base
3. Geographical Base
4. Chronological or Temporal Base
1. Qualitative Base: When the data are classified according to some quality or attributes
such as sex, religion, literacy, intelligence etc...
2. Quantitative Base: When the data are classified by quantitative characteristics like
heights, weights, ages, income etc..
3. Geographical Base: When the data are classified by geographical regions or location, like
states, provinces, cities, countries etc.
4. Chronological or Temporal Base: When the data are classified or arranged by their time
of occurrence, such as years, months, weeks, days etc.... For example, Time Series Data.
TYPES OF CLASSIFICATION
1. One -way classification: If we classify observed data keeping in vi ew single characteristic,
this type of classification is known as one -way classification.
For example, the population of world may be classified by religion as Muslim, Christian etc.
2. Two -way classification: If we consider two characteristics at a time i n order to classify the
observed data then we are doing two -way classification.
For example, the population of world may be classified by Religion and Sex.
3. Multi -way classification: We may consider more than two characteristics at a time to
classify giv en data or observed data. In this way we deal in multi -way classification.
For example, the population of world may be classified by Religion, Sex and Literacy.
TABULATION OF DATA
The process of placing classified data into tabular form is known as tabula tion. A table is a
symmetric arrangement of statistical data in rows and columns. Rows are horizontal
arrangements whereas columns are vertical arrangements. It may be simple, double or
complex depending upon the type of classification.
TYPES OF TABULATIO N
1. Simple Tabulation or One -way tabulation: When the data are tabulated to one
characteristic, it is said to be simple tabulation or one -way tabulation.
For example, tabulation of data on population of world classified by one characteristic like
Religion is example of simple tabulation.
2. Double Tabulation or Two -way tabulation: When the data are tabulated according to
two characteristics at a time. It is said to be double tabulation or two -way tabulation.
For example, tabulation of data on population of world classified by two characteristics like
religion and sex is example of double tabulation.
3. Complex Tabulation: When the data are tabulated according to many characteristics, it is
said to be complex tabulation.
For example, tabulation of data on population of world classified by two characteristics like
Religion, Sex and Literacy etc... is example of complex tabulation.
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DIFFERENCES BETWEEN CLASSIFICATION AND TABULATION
1. First the data are classified and then t hey are presented in tables, the classification and
tabulation in fact goes together. So classification is the basis for tabulation.
2. Tabulation is a mechanical function of classification because in tabulation classified data
are placed in row and columns.
3. Classification is a process of statistical analysis where as tabulation is a process of
presenting the data in suitable form.
FREQUENCY DISTRIBUTION
A frequency distribution is a tabular arrangement of data into classes according to the size o r
magnitude along with corresponding class frequencies (the number of values fall in each
class).
Ungrouped data or Raw Data
Data which have not been arranged in a systemic order is called ungrouped or raw data.
Grouped Data
Data presented in the form of f requency distribution is called grouped data.
Array
The numerical raw data is arranged in ascending or descending order is called an array.
Example
Array the following data in ascending or descending order 6, 4, 13, 7, 10, 16, 19.
Solution
Array in ascendi ng order is 4, 6, 7, 10, 13, 16 and 19.
Array in descending order is 19, 16, 13, 10, 7, 6, and 4.
CLASS LIMITS
The variant values of the classes or groups are called the class limits. The smaller value of the
class is called lower class limit and larger value of the class is called upper class limit. Class
limits are also called inclusive classes.
For example, let us take class 10 -19, the smaller value 10 is lower class limit and larger value
19 is called upper class limit.
CLASS BOUNDARIES
The true valu es, which describes the actual class limits of a class, are called class boundaries.
The smaller true value is called the lower class boundary and the larger true value is called
the upper class boundary of the class. It is important to note that the upper class boundary of
a class coincides with the lower class boundary of the next class. Class boundaries are also
known as exclusive classes.
For example,
Weights in Kg Number of
Students
60-65
65-70
70-75 8
12
5
25 munotes.in
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A student whose weights are between 60 kg and 64.5 kg would be included in the 60 -65
class. A student whose weight is 65 kg would be included in next class 65 -70.
A class has either no lower class limit or no upper class limit in a frequency table is called an
open -end class. We do not like to use open -end classes in practice, because they create
problems in calculation.
For example,
Weights (Pounds) Number of Person s
Below - 110
110-120
120-130
130-140
140-above 6
12
20
10
2
Class Mark or Mid Point
The class marks or mid point is the mean of lower and upper class limits or boundaries. So it
divides the class into two equal parts. It is obtained by dividing the sum of lower and upper -
class limit or class boundaries of a class by 2.
For example, The class mark or mid -point of the class 60 -69 is 60+69/2 = 64.5
Size of Class Interval
The difference between the upper and lower class boundaries (not between class limits) of a
class or the difference betwee n two successive mid points is called size of class interval.
CONSTRUCTIN OF FREQUENCY DIST RIBUTION
Following steps are involved in the construction of a frequency distribution.
1. Find the range of the data: The range is the difference between the largest and the
smallest values.
2. Decide the approximate number of classes: Which the data are to be grouped. There are
no hard and first rules for number of classes. Most of the cases we have 5 to 20 classes. H. A.
Sturges has given a formula for determining the approximation number of classes.
K = 1 + 3.322 log N
where K = Numb er of classes
where log N = Logarithm of the total number of observations
For example, if the total number of observations is 50, the number of classes would be:
K = 1 + 3.322 log N
K = 1 + 3.322 log 50
K = 1 + 3.322 (1.69897)
K = 1 + 5.644
K = 6.644 or 7 classes approximately.
3. Determine the approximate class interval size: The size of class interval is obtained by
dividing the range of data by number of classes and denoted by h class interval size
(h) = Range/Number of Classes
In case of fractional resu lts, the next higher whole number is taken as the size of the class
interval. munotes.in
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4. Decide the starting Point: The lower class limits or class boundary should cover the
smallest value in the raw data. It is a multiple of class interval.
For example, 0, 5, 10, 15, 20 etc... are commonly used.
5. Determine the remaining class limits (boundary): When the lowest class boundary of
the lowest class has been decided, then by adding the class interval size to the lower class
boundary, compute the upper class boundary. The remaining lower and upper class limits
may be determined by adding the class interval size repeatedly till the largest value of the
data is observed in the class.
6. Distribute the data into respective classes: All the observations are marked into
respective classes by using Tally Bars (Tally Marks) methods which is suitable for tabulating
the observations into respective classes. The number of tally bars is counted to get the
frequency against each class. The frequency of all the classes is noted to g et grouped data or
frequency distribution of the data. The total of the frequency columns must be equal to the
number of observations.
Example, Construction of Frequency Distribution
Construct a frequency distribution with suitable class interval size of marks obtained by 50
students of a class are given below:
23, 50, 38, 42, 63, 75, 12, 33, 26, 39, 35, 47, 43, 52, 56, 59, 64, 77, 15, 21, 51, 54, 72, 68, 36,
65, 52, 60, 27, 34, 47, 48, 55, 58, 59, 62, 51, 48, 50, 41, 57, 65, 54, 43, 56, 44, 30, 46, 67, 53 .
Solution
Arrange the marks in ascending order as:
12, 15, 21, 23, 26, 27, 30, 33, 34, 35, 36, 38, 39, 41, 42, 43, 43, 44, 46, 47, 47, 48, 48, 50, 50,
51, 51, 52, 52, 53, 54, 54, 55, 56, 56, 57, 58, 59, 59, 60, 62, 63, 64, 65, 65, 67, 68, 72, 75, 77.
Minimum value = 12; Maximum value = 77
Range = Maximum value - Minimum value = 77 - 12 = 65
Number of classes = 1 + 3.322 log N
= 1 + 3.322 log 50
= 1 + 3.322 (1.69897)
= 1 + 5.64 = 6.64 or 7 approximate
class interval size (h) = Range/No. of classes = 65/7 = 9.3 or 10.
Marks Class
Limits C.L. Tally Marks Number of
Students ƒ Class
Boundary C.B. Class Marks x
10-19
20-29
30-39
40-49
50-59
60-69
70-79 II
IIII
IIII II
IIII IIII
IIII IIII IIII I
IIII III
III 2
4
7
10
16
8
3 9.5-19.5
19.5-29.5
29.5-39.5
39.5-49.5
49.5-59.5
59.5-69.5
69.5-79.5 10 + 19/2 = 14.5
20 + 29/2 = 24.5
30 + 39/2 = 34.5
40 + 49/2 = 44.5
50 + 59/2 = 54.5
60 + 69/2 = 64.5
70 + 79/2 = 74.5
50
Note: For finding the class boundaries, we take half of the difference between lower class
limit of the 2nd class and upper class limit of the 1st class 20 - 19/2 = 1/2 = 0.5 This value is
subtracted from lower class limit and added in upper class limit to get the required class
boundaries.
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Frequency Distribution by Exclusive Method
Class Boundary C.B. Tally Marks Frequency ƒ
10 - 19
20 - 29
30 - 39
40 - 49
50-59
60 - 69
70 - 79 II
IIII
IIII II
IIII IIII
IIII IIII IIII I
IIII III
III 2
4
7
10
16
8
3
50
CUMULATIVE FREQUENCY DISTRIBUTION
The total frequency of all classes less than the upper class boundary of a given class is called
the cumulative frequency of the class. "A table showing the cumulative frequencies is called
a cumulative frequency distribution". There are two types of cumulative frequency
distribution.
Less than cumulative frequency distribution
It is obtained by adding successively the frequencies of all the previous classes including the
class against which it is written. The cumu late is started from the lowest to the highest size.
More than cumulative frequency distribution
It is obtained by finding the cumulative total of frequencies starting from the highest to the
lowest class. The less than cumulative frequency distribution an d more than cumulative
frequency distribution for the frequency distribution given below are:
Less than C.F. More than C.F.
Class Limit ƒ C.B. Marks C.F Marks C.F.
10 - 19 2 9.5 - 19.5 Less than
19.5 2 9.5 or
more 48 + 2 =
50
20 - 29 4 19.5 - 29.5 Less than
29.5 2 + 4 = 6 19.5 or
more 44 + 4 =
48
30 - 39 7 29.5 - 39.5 Less than
39.5 6 + 7 = 13 29.5 or
more 37 + 7 =
44
40 - 49 10 39.5 - 49.5 Less than
49.5 13 + 10 =
23 39.5 or
more 27 + 10 =
37
50 - 59 16 49.5 - 59.5 Less than
59.5 23 + 16 =
39 49.5 or
more 11 + 16 =
27
60 - 69 8 59.5 - 69.5 Less than
69.5 39 + 8 =
47 59.5 or
more 3 + 8 = 11
70 - 79 3 69.5 - 79.5 Less than
79.5 47 + 3 =
50 69.5 or
more 3
DIAGRAMS AND GRAPHS OF STATISTICAL DATA
We have discussed the techniques of classification and tabulation that help us in organising
the collected data in a meaningful fashion. However, this way of presentation of statistical munotes.in
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data dos not always prove to be interesting to a layman. Too many figu res are often confusing
and fail to convey the message effectively.
One of the most effective and interesting alternative way in which a statistical data may be
presented is through diagrams and graphs. There are several ways in which statistical data
may be displayed pictorially such as different types of graphs and diagrams. The commonly
used diagrams and graphs to be discussed in subsequent paragraphs are given as under:
TYPES OF DIAGRAMS/CHARTS
1. Simple Bar Chart
2. Multiple Bar Chart or Cluster Chart
3. Staked Bar Chart or Sub -Divided Bar Chart or Component Bar Chart
a. Simple Component Bar Chart
b. Percentage Component Bar Chart
c. Sub -Divided Rectangular Bar Chart
d. Pie Chart
4. Histogram
5. Frequency Curve and Polygon
6. Lorens Curve
7. Historigr am
1. SIMPLE BAR CHART
A simple bar chart is used to represent data involving only one variable classified on spatial,
quantitative or temporal basis. In simple bar chart, we make bars of equal width but variable
length, i.e. the magnitude of a quantity is represented by the height or length of the bars.
Following steps are undertaken in drawing a simple bar diagram:
Draw two perpendicular lines one horizontally and the other vertically at an
appropriate place of the paper.
Take the basis of classificatio n along horizontal line (X -axis) and the observed
variable along vertical line (Y -axis) or vice versa.
Mark signs of equal breadth for each class and leave equal or not less than half
breadth in between two classes.
Finally mark the values of the given var iable to prepare required bars.
Sample problem: Make a bar graph that represents exotic pet ownership in the United
States. There are 8,000,000 fish, 1,500,000 rabbits, 1,300,000 turtles, 1,000,000 poultry and
900,000 hamsters.
Step 1: Number the Y -axis with the dependent variable. The dependent variable is the one
being tested in an experiment. In this sample question, the study wanted to know how many
pets were in U.S. households. So the number of pets is the dependent variable. The highest
number in th e study is 8,000,000 and the lowest is 1,000,000 so it makes sense to label the Y -
axis from 0 to 8. munotes.in
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Step 2: Draw your bars. The height of the bar should be even with the correct number on the
Y-axis. Don’t forget to label each bar under the x -axis.
Step 3: Label the X -axis with what the bars represent. For this sample problem, label the x -
axis “Pet Types” and then label the Y -axis with what the Y -axis represents: “Number of pets
(per 1,000 households).” Finally, give your graph a name. For this sampl e problem, call the
graph “Pet ownership (per 1,000 households).
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Optional: In the above graph, I chose to write the actual numbers on the bars themselves.
You don’t have to do this, but if you have numbers than don’t fall on a line (i.e. 900,000),
then i t can help make the graph clearer for a viewer.
Tips:
1. Line the numbers up on the lines of the graph paper, not the spaces.
2. Make all your bars the same width.
2. MULTIPLE BAR CHART
By multiple bars diagram two or more sets of inter related data are represented (multiple bar
diagram facilities comparison between more than one phenomena). The technique of simple
bar chart is used to draw this diagram but the difference is that we use different shades,
colours or dots to distinguish between different ph enomena. We use to draw multiple bar
charts if the total of different phenomena is meaningless.
Sample Example
Draw a multiple bar chart to represent the import and export of Pakistan for the years 1982 -
1988.
Years Imports Exports
Rs. (billion) Rs. (billion)
1982 -83 68.15 34.44
1983 -84 76.71 37.33
1984 -85 89.78 37.98
1985 -86 90.95 49.59
1986 -87 92.43 63.35
1987 -88 111.38 78.44
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3. COMPONENT BAR CHART
Sub-divided or component bar chart is used to represent data in which the total magnitude
is divided into different components.
In this diagram, first we make simple bars for each class taking total magnitude in that
class and then divide these simple bars into parts in the ratio of various components. This
type of diagram shows the variation i n different components without each class as well as
between different classes. Sub -divided bar diagram is also known as component bar chart or
staked chart.
Current and Development Expenditure – Pakistan (All figures in Rs. Billion)
Years Current
Expenditure Development
Expenditure Total
Expenditure
1988 -89 153 48 201
1989 -90 166 56 222
1990 -91 196 65 261
1991 -92 230 91 321
1992 -93 272 76 348
1993 -94 294 71 365
1994 -95 346 82 428
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3. b. PERCENTAGE COMPONENT BAR CHART
Sub-divided bar chart may be drawn on percentage basis. to draw sub -divided bar chart on
percentage basis, we express each component as the percentage of its respective total. In
drawing percentage bar chart, bars of length equal to 100 for each class are drawn at first step
and sub -divided in the proportion of the percentage of their component in the second step.
The diagram so obtained is called percentage component bar chart or percentage staked bar
chart. This type of chart is useful to make comparison in components holding the difference
of total constant.
Areas Under Crop Production (1985 -90)
(‘000 hectors)
Year Wheat Rice Others Total
1985 -86 7403 1863 1926 11192
1986 -87 7706 2066 1906 11678
1987 -88 7308 1963 1612 10883
1988 -89 7730 2042 1966 11738
1989 -90 7759 2107 1970 11836
Percentage Areas Under Production
Year Wheat Rice Others Total
1985 -86 66.2% 16.6% 17.2% 100%
1986 -87 66.0 17.7 16.3 100
1987 -88 67.2 18.0 14.8 100
1988 -89 65.9 17.4 16.7 100
1989 -90 65.6 17.8 16.6 100
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3. d. PIE-CHART
Pie chart can be used to compare the relation between the whole and its components. Pie
chart is a circular diagram and the area of the sector of a circle is used in pie chart. Circles are
drawn with radii proportional to the square root of the q uantities because the area of a circle
is 𝐴=2𝜋𝑟2
To construct a pie chart (sector diagram), we draw a circle with radius (square root of the
total). The total angle of the circle is 360°. The angles of each component are calculated by
the formula:
Angle of Sector = 𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑃𝑎𝑟𝑡
𝑇𝑜𝑡𝑎𝑙× 360°
These angles are made in the circle by means of a protractor to show different
components. The arrangement of the sectors is usually anti -clock wise.
Example
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EXERCISES
1. Draw a histogram of the following data:
Weekly Wages 1 - 10 11 - 20 21 - 30 31 - 40 41 - 50
No. of Workers 14 28 36 12 10
2. The following table shows the temperature for the consecutive five days in a particular
week. Draw range graph.
Day M T W Th F
High° C
Low° C 40
25 35
20 50
40 60
55 25
15
3. The following is the distribution of total house hold expenditure (in Rs.) of 202 workers
in a city.
Expenditure in Rs. 100 - 150 150 - 200 200 - 250 250 - 300
No. of Workers 25 40 33 28
Expenditure in Rs. 300 - 350 350 - 400 400 - 450 450 - 500
No. of Workers 30 22 16 8
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Chapter 3
MEASURES OF DISPERSION
Introduction to Measures of Dispersion, Various methods of Dispersion like Range, Mean
Deviation, Variance, Standard Deviation and Coefficient of Variation, Measurement of
Shapes like Skewness and Kurtosis.
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INTRODUCTION TO MEASURE OF DISPERSION
A modern student of statistics is mainly interested in the study of variability and uncertainty.
We live in a changing world. Changes are taking place in every sphere of life. A man of
Statistics does not show much interest in those things which are constant. The total area of
the earth may not be very important to a research minded person but the area under different
crops, areas covered by forests, area covered by residential and comme rcial buildings are
figures of great importance because these figures keep on changing from time to time and
from place to place. Very large number of experts is engaged in the study of changing
phenomenon. Experts working in different countries of the wor ld keep a watch on forces
which are responsible for bringing changes in the fields of human interest. The agricultural,
industrial and mineral production and their transportation from one part to the other parts of
the world are the matters of great intere st to the economists, statisticians and other experts.
The changes in human population, the changes in standard of living, and changes in literacy
rate and the changes in price attract the experts to make detailed studies about them and then
correlate thes e changes with the human life. Thus variability or variation is something
connected with human life and study is very important for mankind.
DISPERSION
The word dispersion has a technical meaning in Statistics. The average measures the centre of
the data. It is one aspect observations. Another feature of the observations is as to how the
observations are spread about the centre. The observation may be close to the centre or they
may be spread away from the centre. If the observation are close to the centre (usually the
arithmetic mean or median), we say that dispersion, scatter or variation is small. If the
observations are spread away from the centre, we say dispersion is large. Suppose we have
three groups of students who have obtained the following marks in a test. The arithmetic
means of the three groups are also given below:
Group A: 46, 48, 50, 52, 54 𝑋̅A = 50
Group B: 30, 40, 50, 60, 70 𝑋̅B = 50
Group C; 40, 50, 60, 70, 80 𝑋̅C = 60
In a group A and B arithmetic means are equal i.e. 𝑋̅A = 𝑋̅B = 50. But in group A the
observations are concentrated on the centre. All students of group A have almost the same
level of performance. We say that there is consistence in the observations in group A. In
group B the mean is 50 but the observations are no t close to the centre. One observation is as
small as 30 and one observation is as large as 70. Thus, there is greater dispersion in group B.
In group C the mean is 60 but the spread of the observations with respect to the centre 60 is
the same as the spre ad of the observations in group B with r espect to their own centre which
is 50. Thus in group B and C the means are different but their dispersion is the same. In group
A and C the means are different and their dispersions are also different. Dispersion is an
important feature of the observations and it is measured with the help of the measures of
dispersion, scatter or variation. The word variability is also used for this idea of dispersion.
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The study of dispersion is very important in statistical data. I f in a certain factory there is
consistence in the wages of workers, the workers will be satisfied. But if workers have high
wages and some have low wages, there will be unrest among the low paid workers and they
might go on strikes and arrange demonstrati ons. If in a certain country some people are very
poor and some are very rich, we say there is economic disparity. It means that dispersion is
large. The idea of dispersion is important in the study of wages of workers, prices of
commodities, standard of l iving of different people, distribution of wealth, distribution of
land among framers and various other fields of life. Some brief definitions of dispersion are:
1. The degree to which numerical data tend to spread about an average value is called the
dispersion or variation of the data.
2. Dispersion or variation may be defined as a statistics signifying the extent of the scattered
items around a measure of central tendency.
3. Dispersion or variation is the measurement of the scattered size of the ite ms of a series
about the average.
For the study of dispersion, we need some measures which show whether the dispersion is
small or large. There are two types of measures of dispersion, which are:
a. Absolute Measure of Dispersion
b. Relative Measure of Dis persion.
ABSOLUTE MEASURE OF DISPERSION
These measures give us an idea about the amount of dispersion in a set of observations. They
give the answers in the same units as the units of the original observations. When the
observations are in kilograms, the absolute measure is also in kilograms. If we have two sets
of observations, we cannot always use the absolute measures to compare their dispersion. We
shall explain later as to when the absolute measures can be used for comparison of
dispersions in two or more than two sets of data. The absolute measures which are commonly
used are:
1. The Range
2. The Quartile Deviation
3. The Mean Deviation
4. The Standard Deviation and Variance
RELATIVE MEASURE OF DISPERSION
These measures are calculated for the comparison of dispersion in two or more than two sets
of observations. These measures are free of the units in which the original data is measured.
If the original data is in dollar or kilometers, we do not use these units with relative measure
of dispersi on. These measures are a sort of ratio and are called coefficients. Each absolute
measure of dispersion can be converted into its relative measure.
Thus, the relative measures of dispersion are:
1. Coefficient of Range or Coefficient of Dispersion.
2. Coef ficient of Quartile Deviation or Quartile Coefficient of Dispersion.
3. Coefficient of Mean Deviation or Mean Deviation of Dispersion.
4. Coefficient of Standard Deviation or Standard Coefficient of Dispersion.
5. Coefficient of Variation ( a special case of Standard Coefficient of Dispersion). munotes.in
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RANGE AND COEFFICIENT OF RANGE
The Range
Range is defined as the difference between the maximum and the minimum observation of
the given data. If X m denotes the maximum observation X o denotes the minimum observation
then the range is defined as Range = X m - Xo.
In case of grouped data, the range is the difference between the upper boundary of the highest
class and the lower boundary of the lowest class. It is also calculated by using the difference
between the mid poi nts of the highest class and the lowest class. it is the simplest measure of
dispersion. It gives a general idea about the total spread of the observations. It does not enjoy
any prominent place in statistical theory. But it has its application and utility in quality
control methods which are used to maintain the quality of the products produced in factories.
The quality of products is to be kept within certain range of values.
The range is based on the two extreme observations. It gives no weight to the ce ntral values
of the data. It is a poor measure of dispersion and does not give a good picture of the overall
spread of the observations with respect to the centre of the observations. Let us consider three
groups of the data which have the same range:
Grou p A: 30, 40, 40, 40, 40, 40, 50
Group B: 30, 30, 30, 40, 50, 50, 50
Group C: 30, 35, 40, 40, 40, 45, 50
In all the three groups the range is 50 - 30 = 20. In group A there is concentration of
observations in the centre. In group B the observations are friendly with the extreme corner
and in group C the observations are almost equally distributed in the interval from 30 to 50.
The range fails to explain these differences in the three groups of data. This defect in range
cannot be removed even if we calcu late the coefficient of range which is a relative measure of
dispersion. If we calculate the range of a sample, we cannot draw any inferences about the
range of the population.
Coefficient of Range
It is relative measure of dispersion and is based on the value of range. It is also called range
coefficient of dispersion. It is defined as:
Coefficient of Range = Xm − Xo
Xm+ Xo.
The range Xm - Xo is standardised by the total X m + X o.
Let us take two sets of observations. Set A contains marks of five students i n Mathematics
out of 25 marks and group B contains marks of the same student in English out of 100 marks.
Set A: 10, 15, 18, 20, 20
Set B: 30, 35, 40, 45, 50
The values of range and coefficient of range are calculated as
Range Coefficient of Range
Set A: (Mathematics)
Set B: (English) 20 - 10 = 10
50 - 30 = 20 20−10
20+10 = 0.33
50−30
50+30 = 0.25
In set A the range is 10 and in set B the range is 20. Apparently it seems as if there is greater
dispersion in set B. But this is not true. The range of 20 in set B is for large observations and
the range of 10 in set A is for small observations. Thus 20 and 10 cannot be compared munotes.in
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directly. Their base is not the same. Marks in Mathematics are out of 25 and marks of English
are out of 100. Thus, it makes no sense to compare 10 with 20. When we convert these two
values into coefficient of range, we see that c oefficient of range for set A is greater than that
of set B. Thus, there is greater dispersion or variation in set A. The marks of students in
English are more stable than their marks in Mathematics.
Example
Following are the wages of 8 workers of a factory. Find the range and coefficient of range.
Wages in ($) 1400, 1450, 1520, 1380, 1485, 1495, 1575, 1440.
Solution
Here Largest Value = X m = 1575 and Smallest Value = X o = 1380
Range = X m - Xo = 1575 - 1380 = 195.
Coefficient of Range = Xm − Xo
Xm+ Xo = 1575 − 1380
1575 + 1380 = 195
2955 = 0.66
Example
The following distribution gives the numbers of houses and the number of persons per house.
Number of Persons 1 2 3 4 5 6 7 8 9 10
Number of Houses 26 113 120 95 60 42 21 14 5 4
Calculate the range and coefficient of range.
Solution
Here Largest Value = X m = 10 and Smallest Value = X o = 1
Range = X m - Xo = 10 - 1 = 9.
Coefficient of Range = Xm − Xo
Xm+ Xo = 10 − 1
10+ 1 = 9
11 = 0.818
Example
Find the range of the weight of the students of a University.
Weights (Kg) 60-62 63-65 66-68 69-71 72-74
Number of Students 5 18 42 27 8
Calculate the range and coefficient of range.
Solution
Weights (Kg) Class Boundaries Mid Value No. of Students
60-62
63-65
66-68
69-71
72-74 59.5 - 62.5
62.5 - 65.5
65.5 - 68.5
68.5 - 71.5
71.5 - 74.5 61
64
67
70
73 5
18
42
27
8
Method 1
Here X m = Upper class boundary of the highest class = 74.5; X o = Lower Class Boundary of
the lowest class = 59.5
Range = X m - Xo = 74.5 - 59.5 = 15 Kilogram .
Coefficient of Range = Xm − Xo
Xm+ Xo = 74.5 − 59.5
74.5 + 59.5 = 15
134 = 0.1119.
Method 2
Here X m = Mid value of the highest class = 73; X o = Mid Value of the lowest class = 61
Range = X m - Xo = 73 - 61 = 12 Kilogram. munotes.in
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Coefficient of Range = Xm − Xo
Xm+ Xo = 73 − 61
73 + 61 = 12
134 = 0.0895.
QUARTILE DEVIATION AND ITS COEFFICIENT
Quartile Deviation
It is based on the lower Quartile Q1 and the upper quartile Q3. The difference Q3 - Q1 is
called the inter quartile range. The difference Q3 - Q1 divided by 2 is called semi -inter-
quartile range or the quartile deviation. Thus
Quartile Deviation (Q.D) = 𝑄3−𝑄1
2. The quartile deviation is a slightly better measure of
absolute dispersion than the range. But it ignores the observation on the tails. If we take
different samples from a population and cal culate their quartile deviations, their values are
quite likely to be sufficiently different. This is called sampling fluctuation. It is not a popular
measure of dispersion. The quartile deviation calculated from the sample data does not help
us to draw an y conclusion (inference) about the quartile deviation in the population.
Coefficient of Quartile Deviation
A relative measure of dispersion based on the quartile deviation is called the coefficient of
quartile deviation. It is defined as
Coefficient of Q uartile Deviation = 𝑄3−𝑄1
2
𝑄3+𝑄1
2 = 𝑄3−𝑄1
𝑄3+𝑄1.
It is pure number free of any units of measurement. It can be sued for comparing the
dispersion in two or more than two sets of data.
Example
The Wheat production (in Kg) of 20 acres is given as: 1120 , 1240, 1320, 1040, 1080, 1200,
1440, 1360, 1680, 1730, 1785, 1342, 1960, 1880, 1755, 1600, 1470, 1750 and 1885. Find the
quartile deviation and coefficient of quartile deviation.
Solution
After arranging the observation in ascending order, we get, 1040, 1080, 1120, 1200, 1240,
1320, 1342, 1360, 1440, 1470, 1600, 1680, 1720, 1730, 1750, 1755, 1785, 1880, 1885, 1960.
Q1 = Value of th item
𝑛+1
4
= Value of
20+1
4 th item
= Value of (5.25)th item
= 5th item + 0.25 (6th item - 5th item) = 1240 + 0.25 (1320 - 1240)
Q1 = 1240 + 20 = 1260
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Q3 = Value of 3 (𝑛+1)
4 th item
= Value o f 3 (20+1)
4 th item
= Value od (15.75) th item
15th item + 0.75 (16th item - 15th item) = 1750 + 0.75 (1755 - 1750)
Q3 = 1750 + 3.75 = 1753.75
Quartile Deviation (QD) = 𝑄3−𝑄1
2 = 1753 .75−1260
2 = 492 .75
2 = 246.875
Coefficient of Quartile Deviation = 𝑄3−𝑄1
𝑄3+𝑄1 = 1753 .75−1260
1753 .75+1260 = 0.164.
Example
Calculate the quartile deviation and coefficient of quartile deviation from the data given
below:
Maximum Load (Short tons) Number of Cables
9.3 - 9.7
9.8 - 10.2
10.3 - 10.7
10.8 - 11.2
11.3 - 11.7
11.8 - 12.2
12.3 - 12.7
12.8 - 13.2 2
5
12
17
14
6
3
1
Solution
The necessary calculations are given below:
Maximum Load
(Short Tons) Number of Cables
F Class Boundaries Cumulative
Frequencies
9.3 - 9.7
9.8 - 10.2
10.3 - 10.7
10.8 - 11.2
11.3 - 11.7
11.8 - 12.2
12.3 - 12.7
12.8 - 13.2 2
5
12
17
14
6
3
1 9.25 - 9.75
9.75 - 10.25
10.25 - 10.75
10.75 - 11.25
11.25 - 11.75
11.75 - 12. 25
12.25 - 12. 75
12. 75 - 13.25 2
2 + 5 = 7
7 + 12 = 19
19 + 17 = 36
36 + 14 = 50
50 + 6 = 56
56 + 3 = 59
59 + 1 = 60
Q1 = Value of [ 𝑛
4] th item
= Value of [60
4] th item
= 15th item
Q1 lies in the class 10.25 - 10.75
⸫ Q1 = 1 + ℎ
𝑓 [𝑛
4− 𝑐]
Where 1 = 10.25, h = 0.5, f = 12, n/4 = 15 and c = 7 munotes.in
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Q1 = 10.25 + 0.25
12 (15 - 7)
= 10.25 + 0.33
= 10.58
Q1 = Value of [ 3𝑛
4] th item
= value of [3 𝑋 60
4] th item
= 45th item
Q3 lies in the class 11.25 - 11.75
⸫ Q3 = 1 + ℎ
𝑓 [3𝑛
4− 𝑐]
where 1 = 11.25, h = 0.5, f = 14, 3n/4 = 45 and c = 36
⸫ Q1 = 11.25 + 0.5
12 (45 - 36)
= 11.25 + 0.32
= 11.57
Quartile Deviation (Q.D) = 𝑄3−𝑄1
2
= 11.57−10.58
2
= 0.99
2 = 0.495
Coefficient of Quartile Deviation = 𝑄3−𝑄1
𝑄3+𝑄1
= 11.57−10.58
11.57+10.58 = 0.99
22.15 = 0.045
THE MEAN DEVIATION
The mean deviation or the average deviation is defined as the mean of the absolute deviations
of observations from some suitable average which may be arithmetic mean, the median or the
mode. The difference (X - average) is called deviati on and when we ignore the negative sign,
this deviation is written as |𝑋−𝑎𝑣𝑒𝑟𝑎𝑔𝑒 | and is read as mod deviations. The mean of these
more or absolute deviations is called the mean deviation or the mean absolute deviation. Thus
for sample data in which the suitable average is the 𝑋̅, the mean deviation (M.D) is given by
the relation
M.D = Ʃ|𝑋−𝑋̅ |
𝑛
For frequency distribution, the mean deviation is given by
M.D = Ʃ𝑓|𝑋−𝑋̅,|
Ʃ𝑓
When the mean deviation is calculated about the median, the formula becomes
M.D. (about median) = Ʃ𝑓|𝑋−𝑀𝑒𝑑𝑖𝑎𝑛 |
Ʃ𝑓
The mean deviation about the mode is
M.D (about mode) = Ʃ𝑓|𝑋−𝑀𝑒𝑑𝑖𝑎𝑛 |
Ʃ𝑓
For a population data the mean deviation about the population mean µ is
M.D = Ʃ𝑓|𝑋−µ |
Ʃ𝑓 munotes.in
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The mean deviation is a better measure of abs olute dispersion than the range and the quartile
deviation.
A drawback in the mean deviation is that we use the absolute deviations |𝑋−𝑎𝑣𝑒𝑟𝑎𝑔𝑒 |
which does not seem logical. The reason for this is that Ʃ (X - 𝑋̅) is always equal to zero.
Even if we use median or more in place of 𝑋̅, even then the summation Ʃ (X - median) or Ʃ
(X - mode) will be zero or approximately zero with the result that the mean deviation would
always be better either zero or close to zero. T hus, the very definition of the mean deviation
is possible only on the absolute deviations.
The mean deviation is based on all the observations, a property which is not possessed by the
range and the quartile deviation. The formula of the mean deviation g ives a mathematical
impression that it is a better way of measuring the variation in the data. Any suitable average
among the mean, median or more can be used in its calculation but the value of the mean
deviation is minimum if the deviations are taken fro m the median. A drawback of the mean
deviation is that it cannot be used in statistical inference.
Coefficient of the Mean Deviation
A relative measure of dispersion based on the mean deviation is called the coefficient of the
mean deviation or the coefficient of dispersion. It is defined as the ratio of the mean deviation
to the average used in the calculation of the mean deviation.
Thus,
Coefficient of M.D (about mean) = Mean Deviation from Mean/Mean
Coefficient of M.D (about median) = Mean Deviati on from Median/Median
Coefficient of M.D (about mode) = Mean Deviation from Mode/Mode
Example
Calculate the mean deviation from (1) Arithmetic Mean (2) Median (3) Mode in respect of
the marks obtained by nine students given below and show that the mean de viation from
median is minimum.
Marks out of 25: 7, 4, 10, 9, 15, 12, 7, 9, 7
Solution
After arranging the observations in ascending order, we get
Marks 4, 7, 7, 7, 9, 9, 10, 12, 15
Mean = Ʃ𝑋
𝑛 = 80
9 = 8.89
Median = Value of (𝑛+1
2) th item
= Value of (9+1
2) th item
= Value of (5) the item = 9
Mode = 7 (Since 7 is repeated maximum number of times)
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Marks X |X − 𝑋̅| |𝑋−𝑚𝑒𝑑𝑖𝑎𝑛 | |𝑋−𝑚𝑜𝑑𝑒 |
4
7
7
7
9
9
10
12
15 4.89
1.89
1.89
1.89
0.11
0.11
1.11
3.11
6.11 5
2
2
2
0
0
1
3
6 3
0
0
0
2
2
3
5
8
Total
21.11 21 23
M.D from mean = Ʃ𝑓|𝑋−𝑋̅ |
𝑛
= 21.11
9 = 2.35
M.D from Median = Ʃ|𝑋−𝑀𝑒𝑑𝑖𝑎𝑛 |
𝑛 = 21
9 = 2.33
M.D from Mode = Ʃ𝑓|𝑋−𝑀𝑜𝑑𝑒 ,|
𝑛 = 23
9 = 2.56
From the above calculations, it is clear that the mean deviation from the median has the least
value.
Example
Calculate the mean deviation from mean and its coefficients from the following data:
Size of items 3 - 4 4 - 5 5 - 6 6 - 7 7 - 8 8 - 9 9 - 10
Frequency 3 7 22 60 85 32 8
Solution
The necessary calculation is given below:
Size of Items X F fX |𝑋− 𝑋̅| f |𝑋− 𝑋̅|
3 - 4
4 - 5
5 - 6
6- 7
7 - 8
8 - 9
9 - 10 3.5
4.5
5.5
6.5
7.5
8.5
9.5 3
7
22
60
85
32
8 10.5
31.5
121.0
390.0
637.5
272.0
76.0 3.59
2.59
1.59
0.59
0.41
1.41
2.41 10.77
18.13
34.98
35.40
34.85
45.12
19.28
Total 217 1538.5 198.53
Mean = 𝑋̅ = Ʃ𝑓𝑋
Ʃ𝑓 1538 .5
217 = 7.09
M.D from Mean = Ʃ|𝑋−𝑋̅|
𝑛= 198 .53
217 = 0.915
Coefficient of M.D (Mean) = 𝑀.𝐷 𝑓𝑟𝑜𝑚 𝑚𝑒𝑎𝑛
𝑀𝑒𝑎𝑛 = 0.915
7.09 = 0.129
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Standard Deviation
The standard deviation is defined as the positive square root of the mean of the square
deviations taken from arithmetic mean of the data.
For the sample data the standard deviation is denoted by S and is defined as
S = √Ʃ (𝑋− 𝑋) 2̅̅̅̅̅̅
𝑛
For a population data the standard deviation is denoted by σ (sigma) and is defined as:
σ = √Ʃ (𝑋− µ)2
𝑁
For frequency distribution the formulas become
S = √Ʃ𝑓 (𝑋− 𝑋) 2̅̅̅̅̅̅
Ʃ𝑓 or σ = √Ʃ𝑓 (𝑋− µ)2
Ʃ𝑓
The standard deviation is in the same units as the units of the original observations. If the
original observations are in grams, the value of the standa rd deviation will also be in grams.
The standard deviation plays a dominating role for the study of variation in the data. It is a
very widely used measure of dispersion. It stands like a tower among measure of dispersion.
As far as the important statistic al tools are concerned, the first important tool is the mean 𝑋̅
and the second important tool is the standard deviation S. It is based on the observations and
is subject to mathematical treatment. It is of great importance for the analysis of data and for
the various statistical inferences.
However, some alternative methods are also available to compute standard deviation. The
alternative methods simplify the computation. Moreover in discussing, these methods we will
confirm ourselves only to sample data b ecause sample data rather than whole population
confront mostly a statistician.
Actual Mean Method
In applying this method first of all we compute arithmetic mean of the given data either
ungroup or grouped data. Then take the deviation from the actual mean. This method is
already is defined above. The following formulas are applied:
For Ungrouped Data For Grouped Data
S = √Ʃ (𝑋− 𝑋)̅̅̅̅
𝑛2 S = √Ʃ𝑓 (𝑋− 𝑋) 2̅̅̅̅̅̅
Ʃ𝑓
This method is also known as direct method.
Assumed Mean Method
a. We use the following formulas to calculate standard deviation:
For Ungrouped Data For Grouped Data
S = √Ʃ𝐷2
𝑛− (Ʃ𝐷
𝑛)2 S = √Ʃ𝑓𝐷2
Ʃ𝑓− (Ʃ𝑓𝐷
Ʃ𝑓)2
where D = X - A and A is any assumed mean other than zero. This method is also known as
short -cut method.
b. If A is considered to be zero then the above formulas are reduced to the following
formulas:
For Ungrouped Data For Grouped Data
S = √Ʃ𝑋2
𝑛 - (Ʃ𝑥
𝑛)2 S = √Ʃ𝑓𝑋2
Ʃ𝑓− (Ʃ𝑓𝑋
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c. If we are in a position to simplify the calculation by taking some common factor or divisor
from the given data the formulas for computing standard deviation are:
For Ungrouped Data For Grouped Data
S = √Ʃ𝑢2
𝑛 - (Ʃ𝑢
𝑛)2 X c S = √Ʃ𝑓𝑢2
Ʃ𝑓− (Ʃ𝑓𝑢
Ʃ𝑓)2 X c or h
Where u = 𝑋−𝐴
ℎ 𝑜𝑟 𝑐 = 𝐷
ℎ 𝑜𝑟 𝑐 ; h = Class Interval and c = Common Divisor. This method is also
called method of step -deviation.
Examples of Standard Deviation
This tutorial is about some examples of standard deviation using all methods which are
discussed in the previous tutorial.
Example
Calculate the standard deviation for the following sample data using all methods: 2, 4, 8, 6,
10 and 12.
Solution:
Method - 1 Actual mean Method
X (X - 𝑿̅)2
2
4
8
6
10
12
ƩX = 42 (2 - 7)2 = 25
(4 - 7)2 = 9
(8 - 7)2 = 1
(6 - 7)2 = 1
(10 - 7)2 = 9
(12 - 7)2 = 25
Ʃ(X - 𝑿̅)2 = 70
𝑋̅ = Ʃ𝑋
𝑛 = 42
6 = 7
S = √Ʃ (𝑋− 𝑋)̅̅̅̅
𝑛2
S = √70
6 = √35
3 = 3.42
Method 2: Taking assumed mean as 6.
X D = (X - 6) D2
2
4
8
6
10
12 - 4
- 2
2
0
4
6 16
4
4
0
16
36
Total ƩD = 6 ƩD2 = 76
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S = √Ʃ𝐷2
𝑛− (Ʃ𝐷
𝑛)2
S = √76
6− (6
6)2 = √70
6 = √35
3 = 3.42
Method 3: Taking Assumed Mean as Zero
X X2
2
4
8
6
10
12 4
16
64
36
100
144
ƩX = 42 Ʃ X2= 364
S = √Ʃ𝑋2
𝑛 - (Ʃ𝑥
𝑛)2
S = √364
6 - (42
𝑛6)2
S = √70
6 = √35
3 = 3.42
Method 4: Taking 2 as common divisor or factor
X u = (X - 4)/2 u2
2
4
8
6
10
12 - 1
0
2
1
3
4 1
0
4
1
9
16
Total Ʃu = 9 Ʃ u2 = 31
S = √Ʃ𝑢2
𝑛 - (Ʃ𝑢
𝑛)2 X c
S = √31
6 - (9
6) 2 X 2
S = √2.92 X 2 = 3.42.
Example
Calculate standard deviation from the following distribution of marks by using all the
methods:
Marks No. of Students
1 - 3
3 - 5
5 - 7
7 - 9 40
30
20
10
Solution
Method 1: Actual mean method
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Marks f X fX (X - 𝑿̅)2 f (X - 𝑿̅)2
1 - 3
3 - 5
5 - 7
7 - 9 40
30
20
10 2
4
6
8 80
120
120
80 4
0
4
16 160
0
80
160
Total 100 400 400
𝑋̅ = Ʃ𝑓𝑋
Ʃ𝑓 = 400
100 = 4
S = √Ʃ𝑓 (𝑋− 𝑋) 2̅̅̅̅̅̅
Ʃ𝑓
S = √400
100 = √4 = 2 Marks
Method 2: Taking assumed mean as 2
Marks f X D = (X - 2) fD fD 2
1 - 3
3 - 5
5 - 7
7 - 9 40
30
20
10 2
4
6
8 0
2
4
6 0
60
80
60 0
120
320
160
Total 100 200 800
S =√Ʃ𝑓𝐷2
Ʃ𝑓− (Ʃ𝑓𝐷
Ʃ𝑓)2
S = √800
100 − (200
100)2
S = √8− 4 = √4 = 2 Marks
Method 3: Using Assumed Mean as Zero
Marks f X fX fX 2
1 - 3
3 - 5
5 - 7
7 - 9 40
30
20
10 2
4
6
8 80
120
120
80 160
480
720
640
Total 100 400 2000
S = √Ʃ𝑓𝑋2
Ʃ𝑓− (Ʃ𝑓𝑋
Ʃ𝑓)2
S = √2000
100− (400
100)2
S = √20− 16 = √4 = 2 marks.
Method 4: By taking 2 as the Common Divisor
Marks f X u = (X - 2)/2 Fu fu 2
1 - 3
3 - 5
5 - 7
7 - 9 40
30
20
10 2
4
6
8 - 2
- 1
0
1
- 80
- 30
0
10 160
30
0
10
Total 100 - 100 200 munotes.in
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S = √Ʃ𝑓𝑢2
Ʃ𝑓− (Ʃ𝑓𝑢
Ʃ𝑓) 2 X h
S = √200
100− (− 100
100) 2 X 2
S = √2− 1 X 2 = √1 X 2 = 2 marks.
Coefficient of Standard Deviation
The standard deviation is the absolute measure of dispersion. Its relative measure is called
standard coefficient of dispersion or coefficient of standard deviation, It is defined as
Coefficient of Standard Deviation = 𝑆
𝑋
Coefficient of Variation
The most important of all the relative measure of dispersion is the coefficient of variation.
This word is variation and not variance. There is no such thing as coefficient of variance. The
coefficient of variation (CV) is defined as
Coefficient of Variation (C.V) = 𝑆
𝑋 x 100
Thus C.V is the value of S when 𝑋̅ is assumed equal to 100. It is a pure number and the unit
of observations is not mentioned with its value. It is written in percentag e form like 20% or
25%. When its value is 20%, it means that when the mean of the observation is assumed
equal to 100, their standard deviation will be 20. The C.V is used to compare the dispersion
in different sets of data particularly the data which diff er in their means or differ in the units
of measurement. The wages of workers may be in dollars and the consumption of meat in
their families may be in kilograms. The standard deviation of wages in dollars cannot be
compared with the standard deviation of amount of meat in kilograms. Both the standard
deviations need to be converted into coefficient of variation for comparison. Suppose the
value of C.V for wages is 10% and the values of C.V for kilograms of meat is 25%. This
means that the wages of workers are consistent. Their wages are close to the overall average
of their wages. But the families consume meat in quite different quantities. Some families use
very small quantities of meat and some others use large quantities of meat. We say that there
is gre ater variation in their consumption of meat. The observations about the quantity of meat
are more dispersed or more variant.
Example
Calculate the coefficient of standard deviation and coefficient of variation for the following
sample data: 2, 4, 8, 6, 10 and 12.
Solution
X (X - 𝑿̅)2
2
4
8
6
10
12 (2 - 7)2 = 25
(4 - 7)2 = 9
(8 - 7)2 = 1
(6 - 7)2 = 1
(10 - 7)2 = 9
(12 - 7)2 = 25
ƩX = 42 Ʃ(X - 𝑿̅)2 = 70 munotes.in
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𝑋̅ = Ʃ𝑋
𝑛 42
6 = 7
S = √Ʃ(X - 𝑋̅)2/n
S = √70
6 = √35
3 = 3.42
Coefficient of Standard Deviation = 𝑆
𝑋̅ = 3.42
7 = 0.49
Coefficient of Variation (C.V) = 𝑆
𝑋̅ X 100 = 3.42
7 X 100 = 48 .86%
USES OF COEFFICIENT OF VARIATION
Coefficient of variation is used to know the consistency of the data. By consistency
we mean the uniformity in the values of the data/distribution from arithmetic mean of
the data/distribution. A distribution with smaller C.V than the other is taken as more
consistent than the other.
C.V is also very useful when comparing two or more sets of data that are measured in
different units of measurement.
THE VARIANCE
Variance is another absolute measure of dispersion. It is defined as the average of the squared
difference between each of the observation in a set of data and the mean. For a sample data
the variance is denoted by S2 and the population variance is denoted by σ2 (sigma square).
The sample variance S2 has the formula
S2 = Ʃ (𝑋− 𝑋)̅̅̅̅
𝑛2
where 𝑋̅ is sample mean and n is the number of observations in the sample.
The population variance σ2 is defined as
σ2 = Ʃ (𝑋− µ)2
𝑁
where µ is the mean of the population and N is the number of observations in the data. It
may be remembered that the population variance σ2 is usually not calculated. The sample
variance S2 is calculated and if need be, this S2 is used to make inference abou t the population
variance.
The term Ʃ (X - 𝑋̅)2 is positive, therefore S2 is always positive. If the original observations are
in centimetre, the value of the variance will be (Centimetre) 2. Thus the unit of S2 is the
square of the units of the original measurement.
For a frequency distribution the sample variance S2 is defined as
S2 = Ʃ𝑓(𝑋− 𝑋)2̅̅̅̅̅
Ʃ𝑓
For a frequency distribution the population variance σ2 is defined as
σ2 = Ʃ 𝑓(𝑋− µ)2
Ʃ 𝑓
In simple words we ca n say that variance is the square root of standard deviation.
Variance = (Standard Deviation)2
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Example
Calculate variance from the following distribution of marks:
Marks No. of Students
1 - 3
3 - 5
5 - 7
7 - 9 40
30
20
10
Solution
Marks F X fX (X - 𝑿)̅̅̅̅2 f (X - 𝑿)̅̅̅̅2
1 -3
3 - 5
5- 7
7 - 9 40
30
20
10 2
4
6
8 80
120
120
80 4
0
4
16 160
0
80
160
Total 100 400 400
𝑋̅ = Ʃ𝑓𝑋
Ʃ𝑓 = 400
100 = 4
S2 = Ʃ𝑓 (𝑋− 𝑋)2̅̅̅̅̅
Ʃ𝑓 = 400
100 = 4
Variance S2 = 4.
SKEWNESS AND KURTOSIS
Skewness is the absence of symmetry in a distribution. Though averages and measures of
dispersion are useful in studying the data, the shape of the frequency curve may also be
equally important to the statistician. If we are studying a certain phenomenon over a period of
time, the average may remain the same, but the structure of the distribution may change. Two
distributions may have identical averages, yet one my tail off towards the higher values and
the other towards the lower values.
To study the distribution we need a measure of this ten dency which will give us the direction
and degree of this tendency which is called skewness.
A fundamental task in many statistical analyses is to characterize
the location and variability of a data set. A further characterization of the data
includes skewness and kurtosis.
Skewness is a measure of symmetry, or more precisely, the lack of symmetr y.
A distribution, or data set, is symmetric if it looks the same to the left and
right of the center point.
Kurtosis is a measure of whether the data are heavy -tailed or light -tailed
relative to a normal distribution. That is, data sets with high kurtosis tend to
have heavy tails, or outliers. Data sets with low kurtosis tend to have light
tails, or lack of outliers. A uniform distribution would be the extreme case.
The histo gram is an effective graphical technique for showing both the
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For univariate data Y1, Y2, ..., YN, the formula for skewness is:
g1=∑Ni=1(Yi−Y¯)3/Ns3
where Y¯ is the mean, s is the standard deviation, and N is the number of data
points. Note that in computing the skewness, the s is computed with N in the
denominator rather than N - 1.
The above formula for skewness is referred to as the Fisher -Pearson
coefficient of skewness. Many software programs actually compute the
adjusted Fisher -Pearson coefficient of skewness
G1=N(N−1)−−−−−−−−√N−2∑Ni=1(Yi−Y¯)3/Ns3
This is an adjustment for sample size. The adjustment approaches 1 as N gets
large. For reference, the adjustment factor is 1.49 for N = 5, 1.19 for N = 10,
1.08 for N = 20, 1.05 for N = 30, and 1.02 for N = 100.
The skewness for a normal distribution is zero, and any symmetric data
should have a skewness near zero. Negative values for the skewn ess indicate
data that are skewed left and positive values for the skewness indicate data
that are skewed right. By skewed left, we mean that the left tail is long
relative to the right tail. Similarly, skewed right means that the right tail is
long relati ve to the left tail. If the data are multi -modal, then this may affect
the sign of the skewness.
Some measurements have a lower bound and are skewed right. For example,
in reliability studies, failure times cannot be negative.
It should be noted that there are alternative definitions of skewness in the
literature. For example, the Galton skewness (also known as Bowley's
skewness) is defined as
Galton skewness=Q1+Q3−2Q2Q3−Q1
where Q1 is the lower quartile, Q3 is the upper quartile, and Q2 is the median.
The Pearson 2 skewness coefficient is defined as
Sk2=3(Y¯−Y~)s
where Y~ is the sample median.
There are many other definitions for skewness that will not be discussed here.
KURTOSIS
For univariate data Y1, Y2, ..., YN, the formula for kurtosis is:
kurtosis=∑ Ni=1(Yi−Y¯)4/Ns4
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where Y¯ is the mean, s is the standard deviation, and N is the number of data
points. Note that in computing the kurtosis, the standard deviation is
computed using N in the denominator rather than N - 1.
The kurtosis for a standard normal distribution is three. For this reason, some
sources use the following definition of kurtosis (often referred to as "excess
kurtosis"):
kurtosis=∑Ni=1(Yi−Y¯)4/Ns4−3
This definition is used so that the standard normal distribution has a kurtosis
of zero. In addition, with the second definition positive kurtosis indicates a
"heavy -tailed" distribution and negative kurtosis indicates a "light tailed"
distribution.
Which defi nition of kurtosis is used is a matter of convention (this handbook
uses the original definition). When using software to compute the sample
kurtosis, you need to be aware of which convention is being followed. Many
sources use the term kurtosis when they are actually computing "excess
kurtosis", so it may not always be clear.
The following example shows histograms for 10,000 random numbers
generated from a normal, a double exponential, a Cauchy, and a Weibull
distribution.
The first histogram is a sample from a normal distribution . The normal
distribution is a symmetric distribution with well -behaved tails. This is
indicated by the skewness of 0.03. The kurtosis of 2.96 is near the expected
value of 3. The histogram verifies the symmetry.
The second histogram is a sample from a double exponential distribution . The
double exponential is a sy mmetric distribution. Compared to the normal, it
has a stronger peak, more rapid decay, and heavier tails. That is, we would
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expect a skewness near zero and a kurtosis higher than 3. The skewness is
0.06 and the kurtosis is 5.9.
The fourth histogram is a sample from a Weibull distribution with shape
parameter 1.5. The Weibull distribution is a skewed distribution with the
amount of skewness depending on the value of the shape parameter. The
degree of decay as we move away from the center also depends on the value
of the shape parameter. For this data set, the skewness is 1.08 and the kurtosis
is 4.46, which indicates moderate skewness and kurtosis.
Many classical statistical tests and intervals depend on normality
assumptions. Significant skewness and kurtosis clearly indicate that data are
not normal. If a data set exhibits significant skewness or kurtosis (as indicated
by a histogram or the numerical measures), what can we d o about it?
One approach is to apply some type of transformation to try to make the data
normal, or more nearly normal. The Box-Cox transformation is a useful
technique for trying to normalize a data set. In particular, taking the log or
square root of a data set is often useful for data that exhibit moderate right
skewness.
Another approach is to use techniques based on distributions other than the
normal. For example, in r eliability studies, the exponential, Weibull, and
lognormal distributions are typically used as a basis for modeling rather than
using the normal distribution. The probabilit y plot correlation coefficient
plot and the probability plot are useful tools for determining a good
distributional model for the data.
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EXERCISES
Q1. Calculate the range and quartile deviation for wages: Also calculate coefficient of
quartile deviation:
Wages 30 - 32 32 - 34 34 - 36 36 - 38 38 - 40 40 - 42 42 - 44
Labourers 12 18 16 14 12 8 6
Hint: Coefficient of Q.D. = 𝑄3−𝑄1
𝑄3+𝑄1 = 0.081
Q2. Calculate the standard deviation from the following:
Marks 10 20 30 40 50 60
No. of
Students 8 12 20 10 7 3
Hint: σ = √√Ʃ𝑓𝑑12
𝑁− (Ʃ𝑓𝑑1
𝑁) 2 X C
= √√109
60− (5
60) 2 X 10 = 13.5
Q3. Find the mean and standard deviation of the following observations:
X: 1 2 4 6 8 9
Transform the above observation such that the mean of transformed observations becomes
double the mean of X, standard deviation remain unchanged.
Hint: Mean = Ʃ𝑋
𝑁 = 30/6 = 5 Let d = X - 5. Then
Ʃd2 = 52. σ = √Ʃ𝑑2
𝑁 −( Ʃ𝑑
𝑁)2 = √52
6 = 2.94.
Q4. E xplain positive and negative skewness with the help of sketches.
Q5. Write short notes on skewness and kurtosis.
munotes.in
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UNIT 6
TESTING OF HYPOTHESIS - ONE SAMPLE
Unit 1
SYLLABUS
Introduction to Hypothesis testing, Hypothesis Testing Procedure, Two tail and One tail of
Hypothesis, Type I and Type II Errors, Concept of t -test and z -test, Hypothesis testing for
Population Proportion.
munotes.in
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INTRODUCTION
Hypothesis testing begins with an assumption, called a Hypothesis, that we make about a
population parameter. A hypothesis is a supposition made as a basis for reasoning. According
to Prof. Morris Hamburg, "A Hypothesis in statistics is simply a quantitative statement about
a population." Palmer O. Johnson has beautifully des cribed hypothesis as "islands in the
uncharted seas of thought to be used as bases for consolidation and recuperation as we
advance into the unknown."
In order to test a hypothesis, we collect sample data, produce sample statistics, and use this
informati on to decide how likely it is that our hypothesized population parameter is correct.
Say that we assume a certain value for a population mean. To test the validity of our
assumption, we gather sample data and determine the difference between the hypothesiz ed
value and the actual value of the sample mean. Then we judge whether the difference is
significant. The smaller the difference, the greater the likelihood that our hypothesized value
for the mean is correct. The larger the difference, the smaller the li kelihood.
Unfortunately, the difference between the hypothesized population parameter and the actual
sample statistic is more often neither so large that we automatically reject our hypothesis nor
so small that we just as quickly accept it. So in hypothes is testing as in most significant rela -
life decisions, clear -cut solutions are the exception, not the rule.
There can be several types of hypotheses. For example, a coin may be tossed 200 times and
we may get heads 80 times and tails 120 times. We may now be interested in testing the
hypothesis that the coin is unbiased. To take another example we may study the average
weight of the 100 students of a particular college and may get the result as 110 lb. We may
now be interested in testing the hypothesis tha t the sample has been drawn from a population
with average weight 115 lb. Similarly, we may be interested in testing the hypothesis that the
variables in the population are uncorrelated.
Suppose a manager of a large shopping mall tells us that the average work efficiency of her
employees is 90%. How can we test the validity of her hypothesis? using the sampling
methods we learnt earlier, we could calculate the efficiency of a sample of her employees. If
we did this and the sample statistic came out to be 9 3%, we would readily accept the
manager's statement. However, if the sample statistic were 46 percent, we would reject her
assumption as untrue. We can interpret both these outcomes, 93 percent and 46 percent, using
our common sense.
Now suppose that our sample statistic reveals an efficiency of 81 percent. This value is
relatively close to 90%. But is it close enough for us to accept the manager's hypothesis?
Whether we accept or reject the manager's hypothesis, we cannot be absolutel y certain that
our decision is correct; therefore, we will have to learn to deal with uncertainty in our
decision making. We cannot accept or reject a hypothesis about a population parameter
simply by intuition. Instead, we need to learn how to decide obje ctively, on the basis of
sample information, whether to accept or reject a hunch. munotes.in
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HYPOTHESIS TESTING
Use a statistic calculated from the sample to test an assertion about the value of a population
parameter.
STEP 1: Determine the sample statistic to be calculated and formulate the hypothesis.
1. The decision about which sample statistic to calculate depends upon the scale used to
measure the variable.
a proportion (π) is calculated for nominal scaled variables.
a me dian (med) is calculated for ordinal scaled variables.
a mean (µ) is calculated for interval or ratio scaled variables.
2. The hypotheses are:
Null Hypothesis (H 0): H0 specifies a value for the population parameter against which the
sample statistic is tes ted. H 0 always includes an equality.
Alternative Hypothesis (H a): Ha specifies a competing value for the population parameter.
Ha
is formulated to reflect the proposition the researcher wants to verify.
includes a non -equality that is mutually exclusive of H0.
is set up for either a one tailed test or a two tailed test.
The decision about using a one tailed vs. two tailed test depends upon the proposition the
researcher wants to verify. For example, if the mean age of the students in this class is tested
against the value 21, the hypotheses could be:
ONE TAILED TEST TWO TAILED TEST
H0: µ = 21 or H0: µ = 21
Ha: µ > 21 or H a: µ > 21 H0: µ = 21
Ha: µ ≠ 21
STEP: 2 Conduct the test.
1. All hypothesis tests take action on H 0. H0 is either rejected or not rejected. When H 0 is
rejected (not rejected), the proposition in H a is verified (not verified).
2. Conducting the test involves deciding if H 0 should be rejected or not to be rejected.
3. There is always a chance a mistake will b e made when H 0 is rejected or not rejected. This
is because the decision is based on information obtained from a sample rather than the entire
target population, i.e., sampling error. Hypothesis tests are designed to control for Type I
error: rejecting a t rue null hypothesis.
4. One approach to deciding if H 0 should be rejected or not rejected is the critical value
approach. The researcher controls the chance of Type I error by setting the test's level of
significance (α). Traditionally, α is set at either .01, .05, or .10.
With the critical value approach:
Rejecting H 0 when the researcher sets α = .01 means the researcher is willing to accept
no more than a 1% chance that a true null hypothesis is being rejected. The results of
a test at the 1% level of sig nificance are highly significant.
Rejecting H 0 when the researcher sets α = .05 means the researcher is willing to
accept no more than a 5% chance that a true null hypothesis is being rejected. The
results of a test at the 5% level of significance are sign ificant. munotes.in
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Rejecting H 0 when the researcher sets α = .10 means the researcher is willing to
accept no more than a 10% chance that a true null hypothesis is being rejected. The
results of a test at the 10% level of significance are marginally significant.
5. An alternative approach to deciding if H 0 should be rejected or not reject is the p -value
approach. The researcher knows precisely the chance of Type I error because the statistical
package calculates the exact probability that a true null hypothesis is be ing rejected. This
exact probability is called the "p -value."
With the p -value approach:
The researcher sets the test's α level based on how much risk of Type I error the
researcher is willing to tolerate. The α level can be set at any value as long as it is less
than or equal to 0.10.
The researcher rejects H 0 if the p -value < α.
The Methods section of a research report that uses the p -value approach should
include a statement about the level that has been set f or α.
Most Statistical packages calculate the p -value for a 2 -tailed test. If you're conducting
a 1-tailed test you must divide p -value by 2 before deciding if it is acceptable.
In SPSS output, the p -value is labelled "Sig(2 -tailed)".
An Interesting Note
Because the p -value precisely measures the test's chances of Type I error, it measures the
exact α. level the test obtains. Consequently:
The p -value is also called the "obtained α. level".
The smaller (larger) the obtained α. level, the more (less) statist ically significant the
results.
STEP 3: State the results of the test as they relate to the problem under study. When H 0 is
rejected, there is sufficient "evidence" in the data to support the assertion made in H a. When
H0 is not rejected, the data do not contain sufficient "evidence" to support the assertion made
in H a.
EXAMPLE RESEARCH PROBLEM
An ongoing concern of University of Wisconsin System administrators is one frequently
expressed by students and their parents: ea rning a degree from a System University takes
longer than the advertised four years. As aspiring UW -L Bus 230 team decides to look into
the problem. Their research is guided by the hypothesis that the problem, at least in part, is
due UW -L students' lack o f commitment. The team reasons that for students to be committed
to graduating "on time" they must average 15 credit hours a semester (the minimum number
needed to graduate in four years), and study hard enough so they won't have to repeat classes.
The tea m hypothesises that UW -L students are averaging fewer than 15 credit hours per
semester, and are studying less than most faculty recommend: two hours per week for each
credit hour attempted. The team interviews 200 randomly selected CBA undergraduates.
Their questionnaire asks:
1. How many credits are you taking this semester?
2. In a typical week, how many hours do you study? munotes.in
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The results of the analysis of these data appear below. Do these data confirm the research
team's hypothesis?
Step 1: Determine the sample statistic to calculate and formulate the hypotheses.
The sample statistic is a mean (µ) because the variable is measured with a ratio scale.
The test is set up as a one -tail test to evaluate the researchers' position that students
are averaging fewe r than 15 credits per semester.
Null Hypothesis H 0: µ credits = 15
Alternative Hypothesis H a: µ credits < 15 1 tailed test divide Sig (2 -tailed) by 2.
Step 2: Conduct the test.
One-Sample Test
Test Value = 15
t
Df Sig.
(2-tailed) Mean
Difference 95% Confidence Interval
of the Difference
Lower Upper
Credits -4.096 199 .000 - .8850 - 1.3111 - .4589
One-Sample Statistics
N Mean Std.
Deviation Std. Error Mean
Credits 200 14.1150 3.0559 .2161
SPSS OUTPUT: Analyse>Compare Means>One Sample t -test:
p-value/2 ≤ .0005/2 = .000 the chance a true null hypothesis is being rejected is
less than .025%.
.005 < .05 reject H 0 because the p -value is less than the α level.
Note: These results are highly significant because the test's obtained α level is almost zero.
Step 3: State the Results
The data contain sufficient evidence to conclude that UW -L students are averaging fewer
than 15 credit hours per semester.
Step 1: Determine the sample statistic to calculate and formulate t he hypotheses.
The sample statistic is a mean (µ) because the variable is measured with a ratio scale.
The test is set up as a one -tail test to evaluate the researchers' position that students
are averaging fewer than 28 hours of studying per week.
Null Hy pothesis H0: µ study = 28
Alternative Hypothesis H a: µ study < 28 1 tailed test divide Sig (2 -tailed) by
2.
Step 2: Conduct the test
Set α = .05
One Sample Statistics
N Mean Std.
Deviation Std.
Error Mean
STUDY 200 20.7000 11.8619 .8388
SPSS OUTPUT: Analyse>Compare Means>One Sample t -test:
One-Sample Test
Test Value = 15
t
Df Sig.
(2-tailed) Mean
Difference 95% Confidence Interval
of the Difference munotes.in
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Lower Upper
Credits -8.703 199 .000 - 7.3000 - 8.9540 - 5.6460
p-value/2 ≤ .0005/2 = .000 the chance a true null hypothesis is being rejected is
less than .025%.
.005 < .05 reject H 0 because the p -value is less than the α level.
Note: These results are highly significant because the test's obtained α level is almost zero.
Step 3: State the results.
The data contain sufficient evidence to conclude that on average UW -L students study fewer
than 28 hours per week.
PRESENTING STATISTICAL RESULTS
The sample estimate for the average number of cr edit hours UW -L students take per semester
is 14.2 (Figure 1). This value is statistically less than 15 (p -value/2 ≤ .00025, Appendix 2,
p.1), the minimum number of credit hours needed per semester to graduate in four years.
Students study an average of 2 0.7 hours per week (Figure 1). This value is statistically less
than 28 (p -value/2 ≤ .00025, Appendix 2, p.2), the number of study hours per week faculty
would recommend for a 14 hour credit load.
DISCUSSING RESEARCH RESULTS
The results indicate that UW -L student behaviour contributes to terms to graduation that
exceed four years. Students average only 14.2 credit hours per semester. This value is
statistically less than 15(p -value/2 ≤ .00025), the minimum number of credit hours per
semester needed to gradua te on time. Also, students study less than the amount most faculty
recommend. Given a 14 credit hour load, faculty recommend that students study 28 hours per
week. The 20.7 hours UW -L students study is statistically less than 28 (p -value/2 ≤ .00025).
While UW-L, students may be brighter than most thereby needing to study less, it is more
likely that the lack of study effort leads to poor classroom performance and a need to retake
some classes. This would extend the number of semester needed to graduate.
EXA MPLE RESEARCH PROBLEM
One objective of the authors of "Alcohol Consumption and College Life" was to evaluate the
UW-L Spring Core Alcohol and Drug Survey finding that "Most UW -L students have 0 -5
drink a week." To do so their questionnaire asked:
During a typical week, how many days per week do you consume alcoholic beverages?
On average, how many drinks do you consume each time you drink?
To do the analysis, the authors multiplied the responses to Q2 and Q3, and used SPSS to
generate a frequency table of t he product, which they labelled Weekly Consumption:
SPSS Output: Analyse > Descriptive Statistics > Frequencies:
Weekly Consumption
Valid Frequency Percent Valid Percent Cumulative
Percent
0
1
2 24
2
7 16.2
1.4
4.7 16.2
1.4
4.7 16.2
17.6
22.3 munotes.in
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3
4
5
6
7
8
9
10
12
14
15
16
18
20
21
24
27
30
33
36
39
40
45
54
60
72
75
120 11
10
7
7
1
10
1
10
8
4
4
7
6
3
1
3
2
6
1
2
2
3
1
1
1
1
1
1 7.4
6.8
4.7
4.7
0.7
6.8
0.7
6.8
5.4
2.7
2.7
4.7
4.1
2.0
.07
.20
1.4
4.1
0.7
1.4
1.4
2.0
0.7
0.7
0.7
0.7
0.7
0.7 7.4
6.8
4.7
4.7
0.7
6.8
0.7
68
5.4
2.7
2.7
4.7
4.1
2.0
0.7
2.0
1.4
4.1
0.7
1.4
1.4
2.0
0.7
0.7
0.7
0.7
0.7
0.7 29.7
36.5
41.2
45.9
46.6
60.8
54.1
60.8
66.2
68.9
71.6
76.4
80.4
82.4
83.1
85.1
86.5
90.5
91.2
92.6
93.9
95.9
96.6
97.3
98.0
98.6
99.3
100.0
Total 148 100.0 100.0
Using the same approach as the Core Study, the authors concluded that most UW -L students
have 0 -8 drinks per week.
EXAMPLE RESEARCH PROBLEM CONTINUED
The authors of "Alcohol Consumption and College Life" wanted to test the hypothesis that
the aver age number of drinks UW -L student's consume was greater than 8.6, the value that
was found in the Core Study.
Step 1: Determine the sample statistic to calculate and formulate the hypotheses.
The sample statistic is a mean (µ) because the variable is measured with a ratio scale.
The test is set up as a one -tail test to evaluate the researchers' position that students
drink more than 8.6 drinks per week.
Null Hypothesis H 0: µ Weekly Consumption = 8.6
Alternative Hypothesis H a: µ Weekly Consumption > 8.6 1 tailed test
divide Sig (2 -tailed) by 2.
Step 2: Conduct the test.
Set α = .05
SPSS OUTPUT: Analyse >Compare Means > One Sample t -test:
One-Sample Test munotes.in
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Test Value = 8.6
T
Df Sig.
(2-tailed) Mean
Difference 95% Confidence
Interval of the
Difference
Lower Upper
Weekly
Consumption 3.179 147 .002 4.31 1.63 6.98
One-Sample Statistics
N Mean Std.
Deviation Std.
Error Mean
Weekly
Consumption 148 12.91 16.48 1.35
p-value/2 = .002/2 = .001 the chance a true null hypothesis is being rejected is
less than -1%.
.001 < .05 reject H 0 because the p -value is less than the α level.
Note: These results are highly significant because the test's obtained α level is almost .001.
Step 3: State the results.
The data contain sufficient evidence to conclude that on average UW -L students are
consuming on average more than 8.6 drinks per week.
PRESENTING STATISTICAL RESULTS
Frequency Percent Valid Percent Cumulative
Percent
Valid 0
1
2
3
4
5
6
7
8
9 24
2
7
11
10
7
7
1
10
1 16.2
1.4
4.7
7.4
6.8
4.7
4.7
.7
6.8
.7 16.2
1.4
4.7
7.4
6.8
4.7
4.7
.7
6.8
.7 16.2
17.6
22.3
29.7
36.5
41.2
45.9
46.6
53.4
54.1
Weekly Consumption
Figure 6
Another hypothesis tested was that most UW -L students consume five or less drinks per
week. According to the cumulative frequency observed, most (53.4%) UW -L students drink
zero to eight alcoholic beverages per week (Figure 6). Furthermore, the sample estimate for
the average number of drinks consumed per week is 12.91. A one sample t -test found this
figure to be statistically larger than 8.6, the mean figure reported in the Core Study (p -value/2
= .001, Appendix B, p age 30).
DISCUSSING RESEARCH RESULTS
there are some stark differences in the findings of this study and those of the Core Study. In
contrast to the Core Study, which concluded that most UW -L students have 0 -5 drinks a
week, this study found that most stud ents have 0 -8 drinks a week. Using the same munotes.in
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methodology as the Core Study, the cumulative frequency for drinks per week exceeded 50%
(53.4%) at eight drinks. Furthermore, statistical evidence exists to estimate of 12.91 is clearly
statistically larger than the value reported in the Core Study. These differences may be the
consequence of how the samples were chosen. This study's sample was randomly chosen
from a list of all UW -L students. The Core's sample was a "modified stratified random
sampling of classe s" in which General Education, junior and senior level classes were
randomly selected in an effort to reflect the proportion of freshmen, sophomore etc., in the
population. All students in attendance at 24 of the 60 classes selected were surveyed. While
this procedure may result in a sample that is a fair representation of the academic classes, the
time of day the surveyed class met may have influenced the results. For example, 7:45 a.m.
classes may be those that students skip most, especially if the preced ing night involved
drinking. A sampling procedure that might miss drinkers would bias the consumption
numbers downward, and lead to the differences in the findings of the two studies.
EXAMPLE RESEARCH PROBLEM
Ted Skemp, a La Crosse area attorney, was star tled when he read the April, 1996 edition of
the ABA Journal. It reported that lawyers' "[clients complained most often about being
ignored.....more than 20% believed their lawyers failed to return phone calls promptly, [and]
more than 20% believed their l awyers did not pay adequate attention to their cases." To make
sure he was treating his clients right, Mr. Skemp commissioned a Bus 230 team to survey his
clients. The research team prepared a questionnaire that included the questions:
1. When you call Mr. Skemp and have to leave a message, does he return your calls
promptly?
0. No 1. Yes
2. Does Mr. Skemp pay adequate attention to your case?
0. No 1. Yes
The team named the variable measured by Q1 "PHONE," and the variable measured by Q2
"ATTENTION."
Output from statistical analysis of these variables appears below. Present the statistical results
and discuss them in terms of whether or not they are favourable for Mr. Skemp.
Statistician with Mr. Skemp
1.00
0.90 .91
0.80
0.70 7.8
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Phone Attention munotes.in
10
TEST OF HYPOTHESIS EXAMPLES ERROR TYPE I & TYPE II
Example 1
The Alpha -Fetoprotein (AFP) Test has both Type I and Type II error possibilities. This
test screens the mother's blood during pregnancy for AFP and determine risk.
Abnormally high or low levels may indicate Down Syndrome.
H0: patient is healthy
Ha: patient is unhealthy
Error Type I (False Positive) is: Test wrongly indicates that patient has a Down Syndrome,
which means that pregnancy must be aborted for no reason.
Error Type II (False Negative) is: Test is negative and the child will be born with m ultiple
anomalies.
Example 2
The Head of the Cartel is trying to uncover the mole from within his crew.
H0: The henchman was not an undercover Miami Dade Police Officer
Ha: The henchman was an undercover Miami Dade Police Officer
Error Type 1 : (False Posit ive)
The head of the Cartel ended up murdering the henchman that was not an undercover Miami
Dade Police Officer. Although the henchman was innocent, he was killed preventing him
from ever flipping and giving the government information.
Error Type 2: (Fals e Negative)
The head of the Cartel interviews a henchman that wan an undercover Miami Dade Police
Officer, but fails to unveil his true identity. Consequently, he continues to allow exposure of
his operation to the undercover Miami Dade Police officer, and further reveals the ins and
outs of his operation, that will eventually bring him to his demise.
Example 3
Airplane mechanic inspects plane for any irregularities or malfunction.
H0: Plane seems to meet all standards of FAA and is ok -ed to fly.
Ha: Plane seems to NOT meet all standards of FAA and is AOG (airplane on the ground).
Error Type 1: (False Positive): Airplane Reverse Thruster is visually fine and operable but
while check testing light indicator states it is not, it is replaced even though thruster was fine
and operable, thus avoiding any accident or problem.
Error Type 2: (False Negative): Airplane Reverse Thruster seems visually to be
malfunctioning but check testing light indicator states it is Fine & Operable, it is NOT
replaced. At land ing a pilot reports a malfunction with the thruster and cannot reduce speed at
landing, plane is involved in accident and many innocent lives are lost.
Example 4
The mechanic inspects the brake pads for the minimum allowable thickness.
H0: Vehicles breaks meet the standard for the minimum allowable thickness.
Ha: Vehicles brakes do not meet the standard for the minimum allowable thickness.
Error Type 1: (False Positive)
The brakes are fine, but the check indicates you need to replace the brake pads; therefo re any
possible problems with brakes are avoided even though the brakes were not worn.
Error Type 2: (False Negative) munotes.in
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The brake pads are worn to beyond the minimum allowable thickness, but the mechanic does
not find anything wrong with them and does not replace them. Consequently, the driver of the
vehicle gets into an accident because she was unable to break effectively and gets into a fatal
accident.
Example 5
During a boxing match, two contenders bump heads. The referee checks the concussion
on one of the boxers.
H0: The boxer is find and able to continue boxing.
Ha: The boxer is injured and must call the bout.
Error Type 1
The boxer is fine and not seriously injured but the referee finds the concussion too severe and
stops the fight.
Error Type 2
The b oxer is seriously injured and the concussion is detrimental to his health, but the referee
does not find the concussion severe, and allows the fight to continue. Due to the severity of
the cut, the boxer faints in mid fight and goes into a coma.
PROCEDURE OF TESTING A HYPOTHESIS
Following are the steps required for testing a hypothesis:
1. Setting up of the hypothesis.
2. Test Statistic
3. Type I & Type II Error
4. Level of Significance
5. Critical Region and Rejection Region
6. Tailed Test Observation
7. Taking a Decision
1. Setting up of the hypothesis: A statistical hypothesis or simply a hypothesis is a tentative
solution logically drawn concerning any parameter or the population.
Generally two hypothesis are set up. They are referred to as,
a) Null Hyp othesis (H 0): A statistical hypothesis which is stated for the purpose of possible
acceptance is called null hypothesis. It is usually referred to by the symbol (H 0). In the words
of FISHER, "Null Hypothesis is the hypothesis which is tested for possible r ejection
under the assumption that it is true."
b) Alternative Hypothesis (Ha): Any hypothesis which is set up as a complementary to the
null hypothesis is called as alternate hypothesis and is denoted by (Ha).
For example, Null Hypothesis and Alternative Hypothesis in the above examples would be as
follows:
i) H0 :µ = µ 0 and H a : µ > µ 0 or µ < µ0.
ii) H0 : There is no difference between the two Drugs A and B.
Or Ha : Drug A is better than Drug B.
Or Ha : Drug A is inferior to Drug B.
Then from the above, it is clear that the null hypothesis indicates no preferential attitude.
Hence a null hypothesis is a hypothesis of no difference. The main problem of the testing of
hypothesis is to accept or reject the null hypothesis. As against the null hypothesis, the munotes.in
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alternative hypothesis specifies a range of the other values that the statistician believes to be
true. Only one alternative hypothesis is tested against the null hypothesis.
2. Test Static: The next step is to compute an appropriate test static which is based upon an
appropriate probability distribution. It is used to test whether the null hypothesis set up
should be accepted or rejected.
3. Type I and Type II Errors: Acceptance or rejection of a hypothesis is based on the result
of the sample information which may not always be consistent with the population. The
decision may be correct in two ways:
Accepting the null hypothesis when it is true.
Rejecting the null hypothesis when it is false.
The decision may be wrong in two ways:
1. Rejecting the null hypothesis when it is true.
2. Accepting the null hypothesis when it is false.
Actual Decision
Accept Reject
H0 is true Correct Decision (No error) Wrong (Type I Error)
H0 is false Wrong Decision (Type II
Error) Correct Decision (No Error)
4. Level of Significance: The next step is the fixation of the level of significance. Level of
significance is the maximum probability of making Type I error. These types of risks should
be kept low as far as possible say at 5% or 1%.
5. Critical region or Rejection Region: Critical region is the region of rejection of the null
hypothesis. It is a region corresponding the value of the sample observations in the sample
space which leads to rejection of the null hypothesis. A single function of the sample
observations can be fixed and we can determine a region or range of values which lead to
rejection of H 0 whenever the value of the function fails in this region.
If the observed set of results has the probability of more than 5% then the di fference between
the sample result and hypothetical parameter is not significant at 5% level i.e. the difference
is due to fluctuations of sampling and H 0 is accepted. It implies that the sample result
supports the hypothesis. Similarly, if the observed se t of results has the probability less than
5% then the difference is significant at 5% level i.e. the difference is not wholly due to
fluctuations of sampling and H 0 is rejected.
6. Tailed test observation: The critical region is represented by the portio n of the area under
the normal curve. The test of hypothesis is confirmed after looking into this table of
hypothesis.
munotes.in
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7. Taking the decision: Lastly the decision should be arrived at as to the accepting or
rejecting the null hypothesis. If the computed v alue of the test static is less than the critical
value as per the table, the hypothesis should be accepted or vice versa.
STANDARD ERROR
The standard deviation of the sampling distribution of a statistic such as mean, median etc. is
known as standard err or.
USES OF STANDARD ERROR
1. S.E. plays a vital role in the large sample theory and is used in testing of hypothesis.
If the difference between the observed and theoretical value of a statistic is greater than 1.96
times the S.E the hypothesis is reject ed at 5% level of significance and say that the difference
is significant at 5% level.
2. The confidence or probable limits within which the population parameter is expected to lie,
can be determined with the help of S.E.
3. It serves as a measure of reliability: As the S.E. increases the deviation of actual values
from the expected one increase. This indicates that the sample is more unreliable.
TESTING OF HYPOTHESIS USING VARIOUS DISTRIBUTION TESTS
1. T-Distribution
W. S. Gosset under the nom de plu me (pen name) of 'student' first found the distribution t =
𝑥− µ̅̅̅̅̅̅̅
𝑠 of R.A. Fisher later on defined t = 𝑥− µ̅̅̅̅̅̅̅
𝑠 √𝑛 correctly by the equation and found its
distribution in 1926.
Using the notation of the previous article, we define a new statistic t by the equation
t = 𝑥− µ̅̅̅̅̅̅̅
𝑠 √𝑛 = 𝑥− µ̅̅̅̅̅̅̅
𝜎𝑠√(𝑛−1) or t = 𝑥− µ̅̅̅̅̅̅̅
𝑠 − √(𝑣+1)
where v = (n - 1) denote the number of degrees of freedom of t.
Then it may be shown that, for samples of size n from a normal population, the distribution of
t is given by
y = 𝑦0
1+𝑡2
𝑣𝑣+1/2
If we choose y 0 so that the total area under the curve is unity; we shall get
y0 = 1
√𝑛𝐵𝑛
2,1/2
We can easily study the form of the t -distribution. Since only even powers of t appear in its
equation it is symmetrical about t = 0 like the normal distribution, but unlike the normal
distribution, it has g 2 > 0 so that it is more peaked than the normal distribution with the same
standard deviation. Also y attain its maximum value at t = 0 so that the mode coincides with
the mean at t = 0. Again the l imiting form of the distribution when y ® ¥ is given by
y = y 0e - 1/2t2
It follows that t is normally distributed for large samples. munotes.in
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USES OF T -DISTRIBUTION
We have seen that if the sample is large, the use is made of the tables of the normal
probability in tegral in interpreting the results of an experiment and on the basis of that to
reject or accept the null hypothesis.
If, however, the sample size n is small, the normal probability tables will no longer be useful.
Following are the uses of t -distribution:
a. To test the significance of the mean of a small random sample from a normal population.
b. To test the significance of the difference between the means of two samples taken from a
normal population.
c. To test the significance of an observed coefficient of correlation including partial and rank
correlations.
d. To test the significance of an observed regression coefficient.
2. z-TABLES OF POINTS AND THE SIGNIFICANCE TEST
We take y 0 so that the total area under the curve given by unity. The probability that we get a
given value z 0 or greater on random sampling will be given by the area to the right of the
ordinate at z 0. Tables for this probability for various values of z are not ava ilable, since this
probability is difficult to evaluate, since it depends upon two numbers v 1 and v 2.
Fisher has prepared tables showing 5% and 1% points of significance for z. Colcord and
Deming have prepared a table of 0.1 % points of significance. Gene rally, these tables are
sufficient to enable us to gause the significance of an observed value of z.
It should be noted that the z -tables given only critical values corresponding to right -tail areas.
Thus 5% points of z imply that the area to the right of the ordinate at the variable z is 0.05. A
similar remark applies to 1% points of z. In other words, 5% and 1% points of z correspond
to 10% and 2% levels of significance respectively.
USES OF z -DISTRIBUTION
1. To test the significance of mean of various samples having two or more than two values.
2. To test the significance of difference between two samples from given population.
3. To test the significance of an observed coefficients based upon the table prepared by
"FISHER" since, the probability is d ifficult to evaluate based upon two numbers.
4. To test the significance on any observed set of values deriving its critical values
corresponding to 5% and 1% of z (since it uses only "Right Tailed Test" for valuing the
significance testing).
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EXERCIS ES
Q1. Write Explanatory Notes on the following:
a. Type I Error
b. Type II Error
c. Procedure for hypothesis testing.
d. t-distribution test
e. z-distribution test
f. Uses of t -test and z -test
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CHAPTER 6
Unit 2
TESTING OF HYPOTHESIS - TWO SAMPLES (Related and Independent)
SYLLABUS: Introduction, Hypothesis testing for difference between Two population means
using z -statistic, Hypothesis testing for difference between Two population means using t -
statistic, Statistical Inferences about the differences between the Means of Two -related
Populations, Hypothesis testing for the difference in Two P opulation Proportions.
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INTRODUCTION
Having discussed the problems relating to sampling of attributes in the previous section, we
now come to the problems of sampling of variables such as height, weight etc. which may
take any value. It shall not, therefore, be possible for us to classify each member of a sample
under one of two heads, success or failure. The values of the variables given by different
trials will spread over a range, which will be unlimited - limited by practical conside rations,
as in the case of weight of people or limited by theoretical considerations as in the case of
correlation coefficient which cannot lie outside the range +1 to - 1.
There are three main objects in studying problems relating to sampling of variables :
i. To compare observation with expectation and to see how far the deviation of one from the
other can be attributed to fluctuations of sampling;
ii. To estimate from samples some characteristic of the parent population, such as the mean of
a variable; an d
iii. To gauge the reliability of our estimates.
DIFFERENCES BETWEEN SMALL AND LARGE SAMPLES
In this section, we shall be studying problems relating to large samples only. Though it is
difficult to draw a clear -cut line of demarcation between large and s mall samples, it is
normally agreed amongst statisticians that a sample is to be recorded as large only if its size
exceeds 30. The tests of significance used for dealing with problems relating to large samples
are different from the ones used for small sa mples for the reasons that the assumptions that
we make in case of large samples do not hold good for small samples. The assumpti ons made
while dealing with problems relating to large samples are:
i. The random sampling distribution of a statistic is approximately normal; and
ii. Values given by the samples are sufficiently close to the population value and can be sued
in its place for calculating the standard error of the estimate.
While testing th e significance of a statistic in case of large samples, the concept of standard
error discussed earlier is used. The following is a list of the formulae for obtaining standard
error for different statistics:
1. Standard Error of Mean
i. When standard devia tion of the population is known
S. E. 𝑋̅ = 𝜎𝑝
√𝑛
where S.E. 𝑋̅ refers to the standard error of the mean
𝜎𝑝 = Standard deviation of the population
n = number of observations in the sample.
ii. When standard deviation of population is not known, we have to use standard deviation of
the sample in calculating standard error of mean. Consequently, the formula for calculating
standard error is
S. E. 𝑋̅ = 𝜎(𝑠𝑎𝑚𝑝𝑙𝑒 )
√𝑛
where σ denotes standard deviation of the sample. munotes.in
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It should be noted that if standar d deviation of both sample as well as population are available
then we should prefer standard deviation of the population for calculating standard error of
mean.
Fiducial limits of population mean:
95% fiducial limits of population mean are
𝑋̅±1.96𝜎
√𝑛
99% fiducial limits of population mean are
𝑋̅±2.58𝜎
√𝑛
2. S.E. of Median or S.E. Med = 1.25331 𝜎
√𝑛
3. S.E. of Quartiles or S.E. = 1.36263 𝜎
√𝑛
4. S.E. of Quartile Deviation or S.E QD = 0.78672 𝜎
√𝑛
5. S.E. of Mean Deviation or S.E. MD = 0.6028 𝜎
√𝑛
6. S.E. of Standard Deviation or S.E. σ = 𝜎
√2𝑛
7. S.E of Regression Estimate of Y on X or S.E xy. = σ x √1− 𝑟2
8. S.E. of Regression Estimate of X on Y or S.E. yx = σ y √1− 𝑟
The following examples will illustrate how standard error of some of the statistics is
calculated:
Examples
1. Calculate standard error of mean from the following data showing the amount paid by 100
firms in Calcutta on the occasion of Durga Puja.
Mid Value (Rs.) 39 49 59 69 79 89 99
No. of firms 2 3 11 20 32 25 7
Solution:
S.E. 𝑋̅ = 𝜎
√𝑛
Calculation of Standard Deviation
Mid-value
m F (m-69)/10
d' fd' fd'2
39
49
59
69
79
89
99 2
3
11
20
32
25
7 -3
-2
-1
0
+1
+2
+3 -6
-6
-11
0
32
50
21 18
12
11
0
32
100
63
N = 100 Ʃfd' = 80 Ʃ fd'2 = 236
σ = √Ʃ fd′2
𝑁 - (Ʃ fd′
𝑁)2 X C = √236
100 - (80
100)2 X 100 = √2.36− 0.64 X 10 = 1.311 X 10 = 13.11
S.E. 𝑋̅ = 13.11
√100 = 13.11
10 = 1.311.
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STANDARD ERROR OF THE DIFFERENCE BETWEEN THE MEANS OF TWO
SAMPLES
i. If two independent random samples with n 1 and n 2 numbers respectively are drawn from
the same population of standard deviation σ, the standard error of the difference between the
sample means is given by the formula:
S.E. of the difference between sample means
= √𝜎21
𝑛1+𝑛2
If σ is unknown, sample stan dard deviation for combined samples must be substituted.
ii. If two random samples with 𝑋1̅̅̅̅, σ1, n1 and 𝑋2̅̅̅̅, σ2, n2 respectively are drawn from different
populations, then S.E. of the different between the means is given by the formula:
= √σ12
n1 + σ22
n2 and where σ 1 and σ 2 are unknown.
S.E. of difference between means
= √𝑆12
n1 + 𝑆22
n2
where S 1 and S 2 represent standard deviation of the two samples.
EXAMPLES
1. Intelligence test on two groups of boys and girls gave the following results:
Mean S.D. N
Girls 75 15 150
Boys 70 20 250
Is there a significant difference in the mean scores obtained by boys and girls?
Solution:
Let us take the hypothesis that there is no significant difference in the mean scored obtained
by boys and girls.
S.E.( 𝑋1̅̅̅̅ - 𝑋2̅̅̅̅) = √σ12
n1 + σ22
n2 where σ 1 = 15, σ 2 = 20, n 1 = 150 and n 2 = 250
Substituting the values
S.E.( 𝑋1̅̅̅̅ - 𝑋2̅̅̅̅) = √(15)2
150 + (20)2
250 = √1.5+ 1.6 = 1.781
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑆.𝐸 = 75−70
1.781 = 2.84
Since the difference is more than 2.58 (1% level of significance) the hypothesis is rejected.
There seems to be a significant difference in the mean score obtained by boys and girls.
STANDARD ERROR OF THE DIFFERENCE BETWEEN TWO STANDARD
DEVIATIONS
In case of two large random samples, each drawn from a n ormally distributed population, the
S.E. of the difference between the standard deviation is given by:
S.E (σ 1 - σ2) = √σ12+ σ22
2𝑛1+2𝑛2
where population standard deviations are not known
S.E (S 1 - S2) = √S12+ S22
2𝑛1+2𝑛2
EXAMPLE munotes.in
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1. Intelligence test of two groups of boys and girls gave the following results:
Girls: Mean = 84, S.D. = 10, n = 121
Boys: Mean = 81, S.D. = 12, n = 81
a. Is the difference is mean scores significant?
b. Is the difference between standard deviations significant?
SOLUTION:
a. Let us take the hypothesis that there is no difference in mean scores.
S.E.( 𝑋1̅̅̅̅ - 𝑋2̅̅̅̅) = √σ12
n1 + σ22
n2 where σ 1 = 10, σ2 = 12, n1 = 121 and n 2 = 81
Substituting the values
S.E.( 𝑋1̅̅̅̅ - 𝑋2̅̅̅̅) = √(10)2
121 + (12)2
81 = √100 /121 + 144 /81= √2.604 = 1.61
Difference of means (84 - 81) = 3
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑆.𝐸 = 3
161 = 1.86
Since the difference is less than 1.96 S.E. (5% level of significance) the given factors support
hypothesis. Hence the difference in mean scores of boys and girls is not significant.
b. Let us take the hypothesis that there is no difference between the standard deviation of the
two samples.
S.E (σ 1 - σ2) = √σ12+ σ22
2𝑛1+2𝑛2 where σ1 = 10, σ2 = 12, n 1 = 121, n 2 = 81
S.E (σ 1 - σ2) = √(10)2
2 𝑋 121 + (12)2
2 𝑋 81 = √100
242 + 144
162 = √1.302 = 1.14
Difference between the two standard deviations - (12 - 10) = 2
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑆.𝐸 = 2
1.14 = 1.75
Since the difference is less than 1.96 S.E. (5% level of significance) the given factors support
hypothesis. Hence the difference in mean scores of boys and girls is not significant.
TWO -SAMPLE Z -TEST FOR COMPARING TWO MEANS
Requirements : Two normally distributed but independent populations, σ is known
Hypothesis test
Formula :
where and are the means of the two samples, Δ is the hypothesized difference between
the population means (0 if testing for equal means), σ 1 and σ 2 are the standard deviations of
the two populations, and n 1and n 2are the sizes of the two samples.
The amount of a certain trace element in blood is known to vary with a standard deviation of
14.1 ppm (parts per million) for male blood donors and 9.5 ppm for female donors. Random
samples of 75 male and 50 female donors yield concentration means of 28 and 33 ppm,
respectively. What is the likelihood that the po pulation means of concentrations of the
element are the same for men and women?
Null hypothesis : H 0: μ 1 = μ 2
or H 0: μ 1 – μ 2= 0
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alternative hypothesis : H a : μ 1 ≠ μ 2
or: H a : μ 1 – μ 2≠ 0
The computed z‐value is negative because the (larger) mean for females was subtracted from
the (smaller) mean for males. But because the hypothesized difference between the
populations is 0, the order of the samples in this computation is arbitrary — could just as
well have been the female sample mean and the male sample mean, in which case z would
be 2.37 instead of –2.37. An extreme z‐score in either tail of the distribution (plus or minus)
will lead to rejection of the null hypothesis of no difference.
The area of the standard normal curve correspondin g to a z‐score of –2.37 is 0.0089. Because
this test is two‐tailed, that figure is doubled to yield a probability of 0.0178 that the
population means are the same. If the test had been conducted at a pre‐specified significance
level of α < 0.05, the null h ypothesis of equal means could be rejected. If the specified
significance level had been the more conservative (more stringent) α < 0.01, however, the
null hypothesis could not be rejected.
In practice, the two‐sample z‐test is not used often, because the two population standard
deviations σ 1 and σ 2 are usually unknown. Instead, sample standard deviations and
the t‐distribution are used.
Inferences About the Difference Between Two Population Means for Paired Data
Paired samples: The sample selected from the first population is related to the
corresponding sample from the second population.
It is important to distinguish independent samples and paired samples. Some examples are
given as follows.
Compare the time that males and females spend watching TV.
Think about the following, then click on the icon to the left to compare your answers.
A. We randomly select 20 males and 20 females and compare the average time they spend
watching TV. Is this an independent sample or paired sample?
B. We randomly select 20 couples and compare the time the husbands and wives spend
watching TV. Is this an independent sample or paired sample?
The paired t-test will be used when handling hypothesis testing for paired data.
The Paired t-Procedure
Assumptions :
1. Paired sa mples
2. The differences of the pairs follow a
normal distribution or the number of pairs
is large (note here that if the number of
pairs is < 30, we need to check whether
the differences are normal, but we do not
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need to check for the normality of each
popul ation )
Hypothesis :
H0:μd=0
Ha:μd≠0
OR
H0:μd=0
Ha:μd<0
OR
H0:μd=0
Ha:μd>0
t-statistic :
Let d = differences between the pairs of data, then d¯ = mean of these differences.
The test statistics is: t∗=d¯−0sd/n√
degrees of freedom = n - 1
where n denotes the number of pairs or the number of differences.
Paired t-interval :
d¯±tα/2 ⋅sdn−−√
Note: sd¯=sdn√ where sd¯ is the standard deviation of the sample differences.
Example: Drinking Water
Trace metals in drinking water affect the flavor and an unusually high concentration can pose
a health hazard. Ten pairs of data were taken measuring zinc concentration in bottom water
and surface water ( zinc_conc.txt ).
Does the data suggest that the true average concentration in the bottom water exceeds that of
surface water?
Location
1 2 3 4 5 6 7 8 9 10
Zinc
concentration in
bottom water .430 .266 .567 .531 .707 .716 .651 .589 .469 .723
Zinc
concentration in
surface water .415 .238 .390 .410 .605 .609 .632 .523 .411 .612
To perform a paired t-test for the previous trace metal example:
Assumptions :
1. Is this a paired sample? - Yes.
2. Is this a large sample? - No.
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3. Since the sample size is not large enough (less than 30), we need to check whether the
differences follow a normal distribution.
In Minitab, we can use Calc > calculator to obtain diff = bottom - surface and then perform a
probability plot on the differences.
Thus, we conclude that the difference may come from a normal distribution.
Step 1 . Set up the hypotheses:
H0:μd=0
Ha:μd>0
where 'd' is defined as the difference of bottom - surface.
Step 2 . Write down the significance level α=0.05 .
Step 3 . What is the critical value and the rejection region?
α=0.05 , df = 9
t0.05=1.833
rejection region: t>1.833
Step 4 . Compute the value of the test statistic:
t∗=d¯sdn√=0.0804 0.052310√ =4.86
Step 5 . Check whether the test statistic falls in the rejection region and determine whether to
reject H o.
t∗=4.86>1.833
reject H0
Step 6 . State the conclusion in words.
At α=0.05 , we conclude that, on average, the bottom zinc concentration is higher than the surface
zinc concentration.
Using Minitab to Perform a Paired t-Test
You can used a paired t-test in Minitab to perform the test. Alternatively, you can perform a 1 -
sample t-test on difference = bottom - surface.
1. Stat > Basic Statistics > Paired t
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2. Click 'Options' to specify the confidence level for the interval and the alternative hypothesis
you want to test. The default null hypothesis is 0.
The Minitab output for paired T for bottom - surface is as follows:
Paired T for bottom - surface
N Mean StDev SE Mean
Bottom 10 0.5649 0.1468 0.0464
Surface 10 0.4845 0.1312 0.0415
Difference 10 0.0804 0.0523 0.0165
95% lower bound for mean difference: 0.0505
T-Test of mean difference = 0 (vs > 0): T -Value = 4.86 P -Value =
0.000
Note: In Minitab, if you choose a lower -tailed or an upper -tailed hypothesis test, an upper or
lower confidence bound will be constructed, respectively, rather than a confidence interval.
Click on the 'Minitab Movie' icon to display a walk through of ' Conducting a
Paired t -Test'.
Using the p-value to draw a conclusion about our example:
p-value = 0.000 < 0.05
Reject H0 and conclude that bottom zinc concentration is higher than surface zinc concentration.
Note: For the zinc concentration problem, if you do not recognize the paired structure, but
mistakenly use the 2 -sample t-test treating them as independent samples, you will not be able to
reject the null hypothesis. This demonstrates t he importance of distinguishing the two types of
samples. Also, it is wise to design an experiment efficiently whenever possible.
What if the assumption of normality is not satisfied? In this case we would use a nonparametric
1-sample test on the differen ce.
HYPOTHESIS TESTING OF THE DIFFERENCE BETWEEN TWO POPULATION
MEANS
B) Hypothesis testing of the difference between two population means
This is a two sample z test which is used to determine if two population means are equal or
unequal. There are three possibilities for formulating hypotheses.
l. : = :
2. : : <
3. : : >
Procedure
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The same procedure is used in three different situations
Sampling is from normally distributed populations with known variances
Sampling from normally distributed populations where population variances are
unknown
o population variances equal
This is with t distributed as Student's t distribution with ( + -2) degrees of
freedom and a pooled variance.
o population variances unequal
When population variances are unequal, a distribution of t' is used in a manner similar to
calculations of confidence intervals i n similar circumstances.
Sampling from populations that are not normally distributed
If both sample sizes are 30 or larger the central limit theorem is in effect. The test statistic is
If the population variances are unknown, the sample variances are used.
Sampling from normally distributed populations with population variances known
Example 7.3.1
Serum uric acid levels
Is there a difference between the means between individuals with Down's syndrome and
normal ind ividuals?
(1) Data
= 4.5 = 12 = 1
= 3.4 = 15 = 1.5
= .05
(2) Assumptions
two independent random samples
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each drawn from a normally distributed population
(3) Hypotheses
: =
:
(4) Test statistic
This is a two sample z test.
(a) Distribution of test statistic
If the assumptions are correct and is true, the test statistic is distributed as the normal
distribution.
(b) Decision rule
With = .05, the critical values of z are -1.96 and +1.96. We reject if z < -1.96 or z >
+1.96.
(5) Calculation of test statistic
(6) Statistical decision
Reject because 2.57 > 1.96.
(7) Conclusion
From these data, it can be concluded that the population means are not equal. A 95%
confidence interval would give the same conclusion.
p = .0102.
Sampling from normally distributed populations with unknown variances
With equal population variances, we can obtain a pooled value from the sample variances.
Example 7.3.2
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Lung destructive index
We wish to know if we may conclude, at the 95% confidence level, that smokers, in general,
have greater lung damage than do non -smokers.
(1) Data
Smokers: = 17.5 = 16 = 4.4752
Non-Smokers: = 12.4 = 9 = 4.8492
= .05
Calculation of Pooled Variance:
(2) Assumptions
independent random samples
normal distribution of the populations
population variances are equal
(3) Hypotheses
:
: >
(4) Test statistic
(a) Distribution of test statistic
If the assumptions are met and is true, the test statistic is distributed as Student's t
distribution with 23 degree s of freedom.
(b) Decision rule
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With = .05 and df = 23, the critical value of t is 1.7139. We reject if t > 1.7139.
(5) Calculation of test statistic
(6) Statistical decision
Reject because 2.6563 > 1.7139.
(7) Conclusion
On the basis of the data, we conclude that > .
Actual values
t = 2.6558
p = .014
Sampling from populations that are not normally distributed
Example 7.3.4
These data were obtained in a study comparing persons with disabilities with persons wi thout
disabilities. A scale known as the Barriers to Health Promotion Activities for Disabled
Persons (BHADP) Scale gave the data. We wish to know if we may conclude, at the 99%
confidence level, that persons with disabilities score higher than persons w ithout disabilities.
(1) Data
Disabled: = 31.83 = 132 = 7.93
Nondisabled: = 25.07 = 137 = 4.80
= .01
(2) Assumptions
independent random samples
(3) Hypotheses
:
: >
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(4) Test statistic
Because of the large samples, the central limit theorem permits calculation of the z score as
opposed to using t. The z score is calculated using the given sample standard deviations.
(a) Distribution of tes t statistic
If the assumptions are correct and is true, the test statistic is approximately normally
distributed
(b) Decision rule
With = .01 and a one tail test, the critical value of z is 2.33. We reject z > 2.33.
(5) Calculation of test statistic
(6) Statistical decision
Reject because 8.42 > 2.33.
(7) Conclusion
On the basis of these data, the average persons with disabilities score higher on the BHADP
test than do the nondisabled persons.
Actual values
z = 8. 42
p = 1.91 x 10 -17
Paired comparisons
Sometimes data comes from nonindependent samples. An example might be testing "before
and after" of cosmetics or consumer products. We could use a single random sample and do
"before and after" tests on each p erson. A hypothesis test based on these data would be
called a paired comparisons test . Since the observations come in pairs, we can study the
difference, d, between the samples. The difference between each pair of measurements is
called di.
Test stati stic
With a population of n pairs of measurements, forming a simple random sample from a
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normally distributed population, the mean of the difference, , is tested using the following
implementation of t.
Paired comparisons
Example 7.4.1
Very -low-calorie diet (VLCD) Treatment
Table gives B (before) and A (after) treatment data for obese female patients in a weight -loss
program.
We calculate di = A-B for each pair of data resulting in negative values meaning that the
participants l ost weight.
We wish to know if we may conclude, at the 95% confidence level, that the treatment is
effective in causing weight reduction in these people.
(1) Data
Values of di are calculated by subtracting each A from each B to give a negative number. On
the TI -83 calculator place the A data in L1 and the B data in L2. Then make L3 = L1 - L2
and the calculator does each calculation automatically.
In Microsoft Excel put the A data in column A and the B data in column B, without using
column heading s so that the first pair of data are on line 1. In cell C1, enter the following
formula: =a1-b1. This calculates the difference, di, for B - A. Then copy the formula down
column C until the rest of the differences are calculated.
n = 9
= .05
(2) A ssumptions
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the observed differences are a simple random sample from a normally distributed
population of differences
(3) Hypotheses
: 0
: < 0 (meaning that the patients lost weight)
(4) Test statistic
The test statistic is t which is calculated as
(a) Distribution of test statistic
The test statistic is distributed as Student's t with 8 degrees of freedom
(b) Decision rule
With = .05 and 8 df the critical value of t is -1.8595. We reject if t < -1.8595.
(5) Calculation of test statistic
(6) Statistical decision
Reject because -12.7395 < -1.8595
p = 6.79 x 10 -7
(7) Conclusion
On the basis of these data, we conclude that the diet program is effective.
Other cons iderations
a confidence interval for can be constructed
z can be used if the variance is known or if the sample is large .
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CAUTION WHILE USING T -TEST
While drawing inferences on the basis of t -test it should be remembered that the
conclusions arrived at on the basis of the 't -test' are justified only if the assumptions
upon which the test is based are true. If the actual distribution is not normally
distributed then, strictly speaking, the t -test is not justified for small samples. If it is not
a random sample, then the assumption that the observations are statistically
independent is not justified and the conclusions based on the t -test may not be corre ct.
The effect of violating the normality assumption is slight when making inference about
means provided that the sampling is fairly large when dealing with small samples.
However, it is a good idea to check the normality assumption, if possible. A review of
similar samples or related research may provide evidence as to whether or not the
population is normally distributed.
LIMITATIONS OF THE TESTS OF SIGNIFICANCE
In testing statistical significance the following points must be noted:
1. They should not be used mechanically: Tests of significance are simply the raw materials
from which to make decisions, not decisions in themselves. There may be situations where
real differences exist but do not produce evidence that they are statistically significant or th e
other way round. In each case it is absolutely necessary to exercise great care before taking a
decision.
2. Conclusions are to be given in terms of probabilities and not certainties: When a test shows
that a difference was statistically significant, it suggests that the observed difference is
probably not due to chance. Thus statements are not made with certainty but with a
knowledge of probability. "Unusual" events do happen once in a while.
3. They do not tell us "why" the difference exists: Though tes ts can indicate that a difference
has statistical significance, they do not tell us why the difference exists. However, they do
suggest the need for further investigation in order to reach definite answers.
4. If we have confidence in a hypothesis it must have support beyond the statistical evidence.
It must have a rational basis. This phrase suggests two conditions: first, the hypothesis must
be 'reasonable' in the sense of concordance with a prior expectation. Secondly, the hypothesis
must fit logically i nto the relevant body of established knowledge.
The above points clearly show that in problems of statistical significance as in other statistical
problems, technique must be combined with good judgement and knowledge of the subject -
matter.
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EXER CISES
Q1. Explain the concept of standard error and discuss its role in the large sample theory.
2. Explain briefly the procedure followed in testing hypothesis.
3. Give some important applications of the t -test and explain how it helps in making business
decisions.
4. What is null hypothesis? How is it different from alternative hypothesis?
5. The mean life of a sample of 10 electric light bulbs was found to be 1, 456 hours with
standard deviation of 423 hours. A second sample of 17 bulbs chosen from a dif ferent batch
showed a mean life of 1, 280 hours with standard deviation of 398 hours. Is there a
significant difference between the means of the two batches?
6. Test the significance of the correlation r = 0.5 from a sample of size 18 against
hypothetical correlation ρ = 0.7.
7. A correlation coefficient of 0.2 is discovered in a sample of 28 pairs of observations. Use
z-test to find out if this is significantly different from zero.
8. How many pairs of observations must be included in a sample in order tha t an observed
correlation coefficient of value 0.42 shall have a calculated value of t greater than 2.72?
9. State the cautions of using t -test.
10. State the limitations of tests of significance.
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