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11

DATA PRESENTATION

Unit Structure

1.0 Objective

1.1 Introduction

1.2 Data Presentation

1.2.1 Data Types

1.2.1.1 Ungrouped Data

1.2.1.2 Grouped Data

1.2.2 Frequency Distribution

1.2.2.1 Types of class Intervals

1.2.3 Graphs and displays

1.2.3.1 Frequency curve

1.2.3.2 Histogram

1.2.3.3 O give curves

1.2.3.4 Stem and Leaf display

1.3 Summary

1.4 Exercise

1.5 List of References

1.0 OBJECTIVE

The learner will be able to understand variuos data types,

understand frequency distributon and be able to plot simple graphs like

Histograms, O give curve to display data. Also stem and leaf type of

display can be learned from this chapter.

1.1 INTRODUCTION

Any Statistical study involves collecting, processing, analysing

data and then reporting informati on from this data.

Statistics is defined as “Statistics is a science that includes the

methods of collecting, organising, presenting, analysing and interpreting

numerical facts and decision taken on that basis”.munotes.in

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21.2 DATA PRESENTATION

1.2.1 DATA TYPES

Data(or Distribution) can be classified as Ungrouped data and Grouped

Data.

Grouped data can be further classified as Discrete and Continous type.

1.2.1.1 Ungrouped Data

In this type, no grouping is done on data and data is available in

the raw form.

Ex 1 : Age of students in a group of five people can be 35, 38, 37, 30 and

35 years

Ex 2 : Scores of six students in a Statistics test can be 4, 6, 8, 3, 2 and 9

marks

1.2.1.2 Grouped Data

In this type data is grouped for some purpose. Grouped data can be

Discrete or Continuous.

Grouped Discrete Data

Number of occurences of each discrete data can be marked as

frequency of that data value in Discrete type of Data Presentation

Ex 3 : The scores of 100 students in a 10 Marks Physics class test can be

grouped as :

Marks 012345678910

Number of students 2361218151316861

Ex 4: The number of students in a degree college in various courses :

Course BCom BMS BScCS BScIT BAF

Number of students 145 98 62 48 80

Grouped Continuous Data

Some suitabl e class intervals are created and data is placed in the

appropriate class.

Ex 5 : The scores of students in a 100 Marks Calculus class test can be

grouped as :

Marks 0-4040-6060-7575-100

Number of students 12 32 28 12

Ex 6: Expenses per month of fa milies in a society are :

Expenses in

Rupees10,000 -

20,00020,000 -

30,00030,000 -

40,000>40000

Number of

families5 12 18 3munotes.in

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3Ex 7 : Time to manufacture an auto assembly is given in hours

Time (in hrs) 1-33-55-77-99-11

Number of assemblies 1 13 15 12 3

1.2.2 FREQUENCY DISTRIBUION

After collecting data, it can be organised in some meaningful form.

The data is thus compressed in systematic manner, for example collected

data can be organised in a tabular form.

Ex 8 : Following data gives marks score d by students in a test of 10

marks. Prepare frequency distribution table.

2, 4, 8, 6, 3, 4, 5, 4, 8, 6, 5, 3, 2, 0, 3, 5, 8, 9, 8, 3.

Solution:

Marks Tally Marks Frequency

0 | 1

1 0

2 || 2

3 |||| 4

4 ||| 3

5 ||| 3

6 || 2

7 0

8 |||| 4

9 | 1

10 0

Data can also be grouped with some suitable class Interval in

frequency table.

1.2.2.1 Types of Class Intervals

Three methods of making class Intervals are :

a) Exclusive method, b) Inclusive method and c) Open end classes.

a) Exlc usive method

The upper limit of a class becomes the lower limit of the next class

in this method.

For example, classes can (10 -20), (20 -30), (30 -40) and so on.

b) Incusive method

In this type the lower limit of a class is kept onemore than the

upper li mit of the previsous class.

For example, classes can be (10 -19), (20 -29), (30 -39) and so on.munotes.in

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4a) Open end classes

In this type, the lower class limit of the first class is not given. Also

the upper limit of the last class may not be given.

For example, c lasses can be (<100), (100 -200), (200 -300), (>300)

1.2.3 GRAPHS

A frequency distribution can be represented by Graphs. Graphs

represent the data pictorically.

Types of Graphs :

a) Frequency curve

b) Histogram

c) O give curve

d) Stem and Leaf display

1.2.3.1 Frequency curve

Ex 9 : Plot Frequency curveMonthJanFebMarAprilMayJuneJulyAugSeptOctNovDecSales(in Lakh)120 135 148 190 212 250 283 312 287 252 313 314

1.2.3.2 Histogram

In this type, each class is represented by a vertical ba r. The bars are

adjacent to each other in Histogram. The areas of the bars are proportional

to the frequencies.munotes.in

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5Ex 10 : Plot Histogram

Number of employees

10000 -20000 25

20000 -30000 15

30000 -40000 30

40000 -50000 10

Solution :

Ex 11 : Plot H istigram and hence find Mode

CI0-55-1010-1515-2020-2525-30

f20 30 40 50 30 20

0

Mode = 15.4 (Ans)

1.2.3.3 O give curves

An O give curve represents the cumulative frequencies for the classes.051015202530munotes.in

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6Ex 12 : Prepare Less than and More than cu mulative frequency table.

Salary Range No. of workers

10000 -20000 125

20000 -30000 134

30000 -40000 150

40000 -50000 85

50000 -60000 15

Solution :

Salary Range No. of workers Less than cf More than cf

10000 -20000 125 125 510

20000 -30000 134 259 385

30000 -40000 150 409 251

40000 -50000 85 494 101

50000 -60000 16 510 16

O give curves are of two types :

a) Less than O give curve and b) More than O give curve

a) Less than O give curve

Ex 13 : Plot Less than Ogive curve

Class Frequency

10-20 12

0-30 24

30-40 43

40-50 38

50-60 22

60-70 11

Solution :

Class Frequency Cumulative

frequency

0-10 0 0

10-20 12 12

20-30 24 36

30-40 43 79

40-50 38 117

50-60 22 139

60-70 11 150munotes.in

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7

Ex 14 : Plot More than Ogive curve

Class Frequency

5-10 25

10-15 30

15-20 35

20-25 38

25-30 22

35-40 11

40-45 5

45-50 4

Solution :

Class Frequency More than

Cumulative

frequency

5-10 25 170

10-15 30 145

15-20 35 115

20-25 38 80

25-30 22 42

35-40 11 20

40-45 5 9

45-50 4 4

50-55 0 0munotes.in

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8Ex 15: PlotLess than O give curve and hence find Median.

CI0-1010-2020-3030-4040-5050-60

f15 32 41 45 28 15

Solution :

CI0-1010-2020-3030-4040-5050-60

f15 32 43 45 28 15

Cf15 47 90 135 163 178

Median = 29, the point of intersection of cf and Rank lines Ans)

Ex 16 : Plot Less than and More than O give curves

Range f

10-20 5

20-30 15

30-40 20

40-50 10

50-60 10

Solution :

Range fLess than cf More than cf

10-20 55 60

20-30 1520 55

30-40 2040 40

40-50 1050 20

50-60 1060 10munotes.in

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9

1.2.3.4 Stem and Leaf display

Stem and Leaf plot shows exact value of individual observation. It

uses ungrouped data.

Steps to draw Stem and Leaf plot :

1) Divide each value of the observation into two parts. One part consisting

of one or more di gits as stem and rest digits as leaf.

2) The stem values are listed on the left of the vertical line and each leaf

value corresponding to the stem is written in horizontal line to the right

of the stem in the increasing order.

3) The stem and the leaf disp lay gives us the ordered data and the shape of

the distribution.

Ex 17 : Display the given data as stem and leaf

42, 53, 65, 63, 61, 77, 47, 56, 74, 60, 64, 68, 45, 55, 57, 82, 42, 35, 39, 51,

65, 55, 33, 76, 70, 50, 52, 54, 45, 46, 25, 36, 59, 63, 83.

Solution :

Stem Leaf

2 5

3 3, 5, 6, 9

4 2, 2, 5, 5, 6, 7, 9

5 0, 1, 2, 3, 3, 4, 5, 5, 6, 7

6 0, 1, 3, 4, 5, 5, 8

7 0, 4, 6, 7

8 2, 3

Comparison of Histogram and Stem and Leaf plot :

1) Stem and Leaf display is simple to plot

2) Data can be easily se en in both stem and Leaf and Histogram.

3) Hsitogram is more suitable for large data set.munotes.in

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101.3 SUMMARY

1) Data can be of ungrouped or grouped (discrete or continuous) type

2) Frequency table gives count of observations of each variable or each

class

3) Fr equency curve gives data trend over period of time

4) Histogram gives pictorial representation of data in each class

5) O give curve plots cumulative frequencies in successice classs

6) Stem and Leaf plot gives more clear picture of individual data

1.4 EX ERCISE

1) Explain various types of distributons with suitable examples for each.

2) Plot frequency curve

Quarter Expenses

(in K)

I 25

II 32

III 35

IV 25

3) Plot Histogram

Class Frequency

0-4 15

4-8 22

8-12 32

12-16 25

16-20 22

4) Plot Less th an O give curve

Class Frequency

10-20 20

20-30 36

30-40 45

40-50 62

50-60 27

60-70 20

5) Plot More than O give curve

Class Frequency

0-20 15

20-40 16

40-60 32

60-80 24

80-100 22

100-120 20munotes.in

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116) Draw stem and leaf plot

22, 25, 28, 32, 35, 21, 4 2, 42, 53, 52, 33, 35, 46, 51, 44, 34, 42, 53

7)Draw stem and leaf plot

15, 22, 26, 35, 24, 21, 25, 30, 35, 38, 24, 26, 26, 29, 32, 38, 27, 33, 35,

24, 25

1.5 LIST OFREFERENCES

1) Probability, Statistics, design of experiments and queuing theory with

applications of Compter Science, S. K. Trivedi, PHI

2) Applied Statistics, S C Gupta, S Chand

munotes.in

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122

MEASURES OF CENTRAL TENDENCY

Unit Structure

2.0 Objective

2.1 Introduction

2.2 Measures of Cetral tendency

2.2.1 Mean

2.2.1.1 Mean of Ungrouped data

2.2.1.2 Meanof Grouped Discrete data

2.2.1.3 Mean of Grouped Continuous data

2.2.1.4 Merits and Demerits of AM

2.2.2 Median

2.2.2.1 Median of Ungrouped data

2.2.2.2 Median of Grouped Discrete data

2.2.2.3 Median of Grouped Continuous data

2.2.2.4 Merits and Demerits of Median

2.2.3 Mode

2.2.3.1 Mode of Ungrouped data

2.2.3.2 Mode of Grouped Discrete data

2.2.3.3 Mode of Grouped Continuous data

2.2.3.4 Merits and Demerits of Mode

2.2.4 Relationship between Mean, Median and Mode

2.3Summary

2.4 Exercise

2.5 List of References

2.0 OBJECTIVE

Learnr will be able to understand concept of Averages. Also

learner will be able to take decision on correct selection of central value

for the given distribution.

2.1 INTRODUCTION

It is require d to convert the given set of data into some form which

can represent the data. Such reduced or compressed form should be easy

to interpret the distribution and also it should allow further algebraic

treatment. Averages are such compact form of the distrib ution. Suchmunotes.in

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13compact form to represent central tendency of the distribution can also be

calles Averages.

Objective of a good measures of central tendency :

1) To condense the data in a single value

2) To enable comprison among various data sets

Requisite so fag o o dM e a s u r eo fS e n t r a lt e n d e n c y:

1) It should be rigidly defined.

2) It should be simple to nderstand and interpret.

3) It should cover all observations in the data set.

4) It should be capable of further algebraic treatment.

5) It should have goo d sampling stability.

6) It should not be undulyaffeted by extreme values.

7) It should be easy to calculate.

2.2 MEASURES OF CENTRAL TENDENCY

Types of Averages :

There are three types of Averages : Mean, Median and Mode. Also

there are some more types l ike Geometric Mean, Harmonic Mean and

Quantiles.

2.2.1 MEAN

2.2.1.1 Mean of Ungrouped Data

)

For Ungrouped Data :

This can also be written as :

Ex 1 : Find Arithmetic Mean of 4, 5, 2, 5, 7

Solution :

(Ans)

2.2.1.2 Mean of Grouped (Discrete) Data

)munotes.in

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14For Grouped (discrete) Data :

This can also be written as :

Ex 2 : Find Arithmetic Mean (AM) of

X12345

f2012252330

Solution :

X f fX

1 20 20

2 12 24

3 25 75

4 23 92

5 30 150

Total110 361

Mean,

(Ans)

Ex 3 : Marks obtained by students of Discrete mathematics class are as

given below. Find AM.

Marks 12345678910

No of students 1225233023241226133

Solution :

Marks, X 1234 5 6 78 9 10Total

No of

students, f12252330 23 24 1827 14 3191

fX 125069120 115 144 84208 117 30949

Mean,

(Ans)munotes.in

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152.2.1.3 Mean of Grouped (Continuous) Data

)

For Grouped (continuous) Data :

This can also be written as :

Ex 4 : Find Arithmetic Mean (AM) of

Class

Interval15-

2020-

2525-

3030-

3535-

4040-

4545-

5050-

5555-

60

f 4 5 11 6 5 8 9 6 4

Solution :

Class

Interval15-

2020-

2525-

3030-

3535-

4040-

4545-

5050-

5555-

60

f 4 5 11 6 5 8 9 6 4

Class

Mark, X17.522.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5

fX 70 112.5 302.5 195 187.5 340 427.5 315 230

Mean,

(Ans)

Ex 5 : Find Arithmetic Mean (AM) of

Class Interval 10-2020-3030-4040-5050-60

f 15 12 18 19 21

Solution :

Class Interval 10-2020-3030-4040-5050-60Total

f 15 12 18 19 21 85

Class Mark, x 15 25 35 45 55

fX 225 300 630 855 1155 3165

Mean,

(Ans)munotes.in

## Page 16

162.2.1.4 Merits and Demerits of AM

Merits of AM

(i) It is rigidly defined

(ii) It is easy to calculate and easy to understand

(iii) It is based on all observations

(iv) It is capable of further algebraic treatment

Demerits of AM

(i) It is affected by extreme values

(ii) It is not possible to calculate AM for open end class intervals

(iii) It is unduly af fected by extreme values

(iv) It may be number which itself may not be present in data

2.2.2 MEDIAN

2.2.2.1 Median of Ungrouped Data

Median is the positional average of the data set.

Data needs to be arranged in ascending order to find th e Median.

Median is middle value when there are odd number of observations.

Median is average of middle two values when there are even number of

observations.

Ex 6 :F i n dM e d i a no f5 ,4 ,3 ,6 ,8 ,2 ,5

Solution : Arrange the data in ascending order.

2, 3, 4,5,5 ,6 ,8

Median = 5 (Ans)

Ex 7 :F i n dM e d i a no f2 ,4 ,3 ,6 ,8 ,2 ,5 ,6

Solution : Arrange the data in ascending order.

2, 2, 3, 4,5,6, 6, 8

Median =

(Ans)

2.2.2.2 Median of Grouped(discrete) Data

Use cumulative frequ ency to find Median of Grouped(discrete) data.

Ex 8 :F i n dM e d i a n

X 12345

f 2012252330

Solution :

X 12345

f 2012252330

Cf 20327598128munotes.in

## Page 17

17N=1 2 8

Rank = (N+1)/2 = 129/2 = 64.5

Cf value first exceed Rank at 75. So, corresponding X va lue is Median

Median = 3 (Ans)

2.2.2.3 Median of Grouped(continuous) Data

Use cumulative frequency to find Median of Grouped(continuous) data.

Steps :

1) Arrange data in ascending order

2) Obtain cumulative frequency against each class

3)Find sum of all frequencies (N).

4) Find Rank, R=N/2

5) Locate a cumulative frequency which first appears higher than Rank

6) Use given formula to find Median

Where,

Ex 9 :F i n dM e d i a n

Class Interval 0-1010-2020-3030-4040-50

F 2 12 25 23 3

Solution :

Class Interval 0-1010-2020-3030-4040-50

f 2 12 25 23 3

Cf 2 14 39 62 65

(Ans)munotes.in

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18Ex 10 : Find Median

Class Interval 10-

2020-

3030-

4040-

5050-

6060-

7070-

8080-

90

F 16 21 20 28 10 3 1 1

Solution :

Class Inter val 10-

2020-

3030-

4040-

5050-

6060-

7070-

8080-

90Total

f 16 21 20 28 10 3 1 1 100

Cf 16 37 57 85 95 98 99 100

(Ans)

2.2.2.4 Merits and Demerits of MEDIAN

Merits of Median

(i) It is not affected by extreme value

(ii) It is e asy to calculate. Sometimes, Median can be found out simply by

observation

(iii) It can be located Graphically

(iv) It is easy to understand and easy to calulate

Demerits of Median

(i) It does not include all data in the data set

(ii) For larger data sets , arranging numbers in ascending order is tedious

(iii) It is not capable of further algebraic treatment

(iv) It does not capture small changes in data set

2.2.3MODE

Mode is the highest occuring number in the distribution, or it is the

number with the hig hest frquency.

2.2.3.1 Mode of Ungrouped Data

)

Mode of ungrouped data can be simply obtained by observation.

Arrange all the numbers in the ascending (or descending) order and count

the occurrence of each number. The number with thehighest or most

occurrence is Mode. There can bemore than Mode in the distribution.

Ex 11 : Find Mode of 7, 5, 8, 7, 6, 8, 2, 7munotes.in

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19Solution : Arranging inascending order : 2, 8, 6, 7, 7, 7, 8, 8

Since number 7 occurred highest number of times, i.e. three times,

Mode = 7 (Ans)

Ex 12 : Find Mode of 7, 5, 8, 7, 6, 8, 2, 7, 8

Solution : Arranging inascending order : 2, 8, 6, 7, 7, 7, 8, 8, 8

Two numbres 7 and 8 bith occurred three times,

Mode = 7 and Mode = 8 (Ans)

2.2.3.2 Mode of Grouped (discrete) Data

)

Ex 13 : Find Mode

X2345678

F12252863545317

Since highest frequency is 63, corresponding X value is Mode.

Mode = 5 (Ans)

2.2.3.3 Mode of Grouped (continuous) Data

)

Following formula is to be used to find Mode of groupe d

(continuous) data.

Where,

Ex 14 : Find Mode

Range 0-44-88-1212-1616-20

F 12 25 28 63 54

Since hhighest frequency is 63, class interval [12 -16] is Modal class.

Mode = 15.18 (Ans)

Ex 15 : Find Mode

Range 0-1010-2020-3030-40

F 12 25 28 63

Since highest frequency is 63, class interval [30 -40] is Modal class.munotes.in

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20

Mode = 33.57 (Ans)

2.2.3.4 Merits and Demerits of MODE

Merits of Mode

(i) It is not affected by extreme value

(ii) It is easy to calculate. Sometimes, Mode ca nb ef o u n do u ts i m p l yb y

observation

(iii) It can be located Graphically

(iv) It is easy to understand and easy to calulate

Demerits of Mode

(i) It does not include all data in the data set

(ii) Mode is not unique, hence not suitable for further algebraic treatment.

(iii) It does not capture small changes in data set

Ex 16 : The following are the weights of 30 wooden logs :

132, 166, 134, 119, 151, 114, 138, 124, 130, 132,

142, 121, 144, 147, 126, 104, 143, 129, 108, 111,

155, 131, 157, 137, 145, 122, 148, 139, 135, 136.

Arrange the data in a frequency table with class interval of 10 kg. each.

The first interval being 100 -110. Find Arithmetic Mean (AM), Median and

Mode.

Solution :

Class

IntervalMid

value

(X)Tally

markFrequency

(f)fX Cumulative

Frequen cy

(cf)

100-110 105 || 2 210 2

110-120 115 ||| 3 345 5

120-130 125 |||| 5 625 10

130-140 135 |||||||| 10 1350 20

140-150 145 |||| | 6 870 26

150-160 155 ||| 3 465 29

160-170 165 | 1 165 30munotes.in

## Page 21

21Mean :

Mean,

(Ans 1)

Median :

(Ans 2)

Mode :

Mode = 135.56 (Ans 3)

2.2.4 RELATIONSHIP BETWEEN MEAN, MEDIAN AND MODE

For moderately assymetrical distributions, the empirical formula

relating Mean, Median and Mode is :

Ex 17 : Find Mode if Mean is 12 and Median is 15

Solution :

(Ans)

2.3 SUMMARY

Averages (Mean, Median and Mode) represent the central value in

the distribution. The formula for central value depends upon the type of

data. Different data sets can be compar ed using averages of each data set.munotes.in

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222.4 EXERCISE

1) Find AM of 5, 3, 2, 12, 5, 6, 9

2) Find AM of

Class Interval 0-1010-2020-3030-4040-50

f 125 123 234 220 101

3) Find Median class interval from the following distribution

X 200-202 202-204 204-206206-208 208-210

f 145 320 445 469 342

4) Find Median

X 10 12 14 16 18

f 210 223 245 268 213

5) Find Median

X 0-44-88-1212-1616-20

F 65 56 43 69 34

6) Find Mode

X 67891011

F 212325372115

7) Find Mode

Range 0-100 100-200 200-300 300-400 400-500

F 123 145 180 162 121

8) Find Mode if Median is 54 and Mean is 62

2.5 LIST OFREFERENCES

1) Probability, Statistics, design of experiments and queuing theory with

applications of Compter Science, S. K. Trivedi, PHI

2) Applied Statistic s, S C Gupta, S Chand

munotes.in

## Page 23

233

MEASURES OF DISPERSION

Unit Structure

3.0 Objective

3.1 Introduction

3.2 Measures of Dispersion

3.2.1 Variance

3.2.1.1 Variance of Ungrouped data

3.2.1.2 Variance of Grouped Discrete data

3.2.1.3 Variance of Grouped Continuous data

3.2.2 Standard Deviation

3.2.2.1 Standard Deviation of Ungrouped data

3.2.2.2 Standard Deviation of Grouped Discrete data

3.2.2.3 Standard Deviation of Grouped Continuous data

3.2.2.4 Combined Mean and combined standard Deviation

3.2.3 Co efficient of Variation (CoV)

3.2.4 Quartiles

3.3 Summary

3.4 Exercise

3.5 List of References

3.0 OBJECTIVE

The understanding of Dispersion (or deviation) is essential to

completely understand and anlyse the distribution alongwith Cen tral

Tendencies. Variance, Standard Deviation and Quantiles sare useful in

Data analysis. This unit helsp learner to analyse distribution using

measures of deviations.

3.1 INTRODUCTION

The central value of the data can be represented by Averages, the

spread of data can be exlained with the help of Measure of Dispersion.

3.2 MEASURES OF DISPERSIONS

Measure of Dispersion serve the objective of determining the

reliability of an average and compare the variability of different

distributions.munotes.in

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24Requisite of a Good Measure of Dispersion :

1)It should b erogodly defined.

2)It should covr all observations in the distribution

3)It should have Sampling stability

4)It shuld be capable of further Mathematical treatment

5)It should not be duly affected by extreme values

Some i mportant Measures of Disersion are :

1)Variance (v)

2)Standard Deviation (SD)

3)Quartile Deviation (QD)

4)Range

3.2.1 Variance

The Arithmetic Mean of squares of deviations taken from

Arithmetic Mean is called Variance .

3.2.1.1 Variance of Ungrouped data

Alternate and more convinient formula for Variance is,

,

Ex 1 : Find Variance of 3, 6, 8, 1, 3

Solution :

(Ans)

3.2.1.2 Variance of Grouped (discrete) data

Alternate and more convinient formul a for Variance is,

,w h e r e ,

Ex 2 : Find Variance of

X4567

F12242318munotes.in

## Page 25

25Solution :

X45 6 7 Total

F1224 23 18 77

1625 36 49 -

Fx 48120 138 126 432

(Ans)

3.2.1.3Variance of Grou ped (continuous) data

Alternate and more convinient formula for Variance is,

,w h e r e ,

Ex 3 : Find Variance of

X0-44-88-1212-16

F12 24 23 18

Solution :

X 0-44-88-12 12-16Total

f 12 24 23 18 77

X 2 6 10 14 -

4 36 100 196 -

Fx 48 120 138 126 650

48 864 2300 3528 6740

(Ans)

3.2.2 Standard Deviation

Standard Deviation is square root of the variance. One can find

variance and then take square root of variance, which will give stan dard

deviation

3.2.2.1 Standard Deviation of Ungrouped data

Ex 4 : Find standard deviation of 3, 6, 8, 1, 3munotes.in

## Page 26

26Solution :

(Ans)

Ex 5 : Find standard deviation of 49, 63, 46, 59, 65, 52, 60, 54

(Ans)

3.2.2.2 Standard Deviation of Grouped (discrete) data

Standard deviation of Grouped (discrete) data can be found out by

taking square root of variance

Ex 6 : Find Standard Deviation

X23456789

f23425321

Solution :

23456 7 8 9Total

23425 3 2 122

49161030 21 16 9115

8276450180 147 128 81685

(Ans)munotes.in

## Page 27

273.2.2.3 Standard Deviation of Grouped (continuous) data

Standard deviation of Grouped (continuous) data can be foun do u t

by taking square root of variance

Ex 7 : Find standard deviation

X 0-10 10-

2020-

3030-

4040-

5050-

6060-

7070-

80Total

F 2 5 3 6 4 2 1 1

fX

3.2.2.4Combined Mean and combined Standard Deviation

Combined Mean :

Combined Mean of two d ata sets can be found out using following

formula.

Ex 8 : Find combined Mean of following data sets.

Set 1 Set 2

Number of observations 25 45

Mean 8 9

Solution :

(Ans)

Ex 9 : Find Combined Mean

Set 1 Set 2 Set 3

Number of obse rvations 120 135 145

Mean 51 48 46

Solution :

(Ans)munotes.in

## Page 28

28Combined Standard Deviation :

Where,

Ex 10 : Find Combined Mean and Combined Standard Deviation :

Group 1 Group 2

No. of observations 32 25

Mean 12 14

SD 3 4

Solution :

Group 1 Group 2

No. of observations

Mean

SD

(Ans)

3.2.3 Coefficient of Variation (CV)

The Coefficient of Variation is the ratio of standard deviation to

the arithmetic mean express ed as percentage.

CV can be used to know the consistency of the data. A distribution

with smaller CV is more consistent than the other one. CV is also useful

for comparing two or more sets of data that are measued in different units

of measurement.munotes.in

## Page 29

29Ex11 :Find coefficient of variation of 2, 5, 4, 1 and 3

Solution :

x1 0 0= 47% (Ans)

3.2.4 Quartile Deviation (QD)

Quartile Deviation is defined as ,

Where, Q3 is upper (third) quartil and Q1 is lower (fi rst) quartile.

is defined as,

,w h e r e

for

Coefficient of QD is defined as,

Ex 12 : Find QD

Class Interval 0-1010-2020-3030-4040-50

f 2 12 25 23 3

Solution :

Class Interval 0-1010-2020-3030-4040-50

f 2 12 25 23 3

Cf 2 14 39 62 65

To find Q3 :

Select cumulative frequency value higher or equal to Rank,

munotes.in

## Page 30

30To find Q1 :

Select cumulative frequency value higher or equal to Rank,

(Ans)

Ex 13 : Find Co -efficient of QD

Class Interval 0-22-44-66-88-10

f 14 18 21 20 12

Solution :

Class Interval 0-22-44-66-88-10

f 14 18 21 20 12

Cf 14 32 53 73 85

To find Q3 :

Select cumulative frequency value higher or equal to Rank,

To find Q1 :

Select cumulative frequency value higher or equal to Rank,

munotes.in

## Page 31

31

(Ans)

Merits and Demerits of QD

Merits of QD :

1)It is rigidly defined

2)It is not affected by extreme values

3)It can be calculated with open end class intervals

Demerits of QD :

1)It is not based on alll observations

2)It is much affected by sampling fluctuations

3.3 SUMMARY

1) Standard Deviation and Variance are two important measures of

Dispersion.

2) Co efficient of Variation is the ration of standard deviation to mean

expressed as percentage.

3.4 EXERCISE

1) Find SD of 4, 6, 2, 8, 2

2) Find Variance of

X234 56

F6578110 8886

3) Find Standard Deviation of

Range 10-2020-3030-4040-5050-6060-7070-80

F 5 4 8 9 4 5 3

4) Find QD and Coefficient of QD of

Range 0-44-88-1212-1616-2020-2424-28

F 5 12 24 18 16 12 1munotes.in

## Page 32

325) Find Combined Mean and Combined Standard Deviation

Group 1 Group 2 Group 3

No. of observations 120 135 130

Mean 13 16 15

SD 3 5 4

3.5 LIST OFREFERENCES

1) Probability, Statistics, design of experiments and queuing theory with

applications of Compter Science, S. K. Trivedi, PHI

2) Applied Statistics, S C Gupta, S Chand

munotes.in

## Page 33

334

MOMENTS, SKEWNESS AND KU RTOSIS

Unit Structure

4.0 Objective

4.1 Introduction

4.2 Moments

4.3 Relation between Central moments and Raw moments

4.4Skewness

4.5 Kurtosis

4.6 Summary

4.7 Exercise

4.8 List of References

4.0 OBJECTIVE

Moments are used to describe characteristics of a distribution such

as central tendency, dispersion. Skewness refers to the lack of symmetry

of the curve on both sides, whereas, Kurtosis referes to peakedness of the

normal distribution curve.

4.1 INTRODUCTION

Moments are a family of equations, each r epresenting a different

quantity.

Skewness refers to lack of symmetry in the distribution, whereas

Kurtosis refers to peakedness of the normal distribution curve.

Skewness is represented by either Karl Pearson’s measure or Bowley’s

measure of Skewness.

4.2MOMENTS

Moments can be defined as arithmetic mean of different powers of

deviations of observations from a particular value. When that particlular

value is zero, moment is called raw moment , and when that value is

mean, moment is called central moment .munotes.in

## Page 34

34For ungrouped data :

Central Moment for ungrouped data is given as :

In general Moment around a point a is given as

For Grouped data :

Central Moment for grouped data is given as :

In general Moment around a point a is given as

Ex 1: Find first four raw moments of following data :

X2345

f12151815

Solution :

XffX

21224 48 96 192

31545 135 405 1215

41872 288 1152 4608

51575 375 1875 9375

Total 1460216 846 3528 15390

First Raw Moment :

munotes.in

## Page 35

35Second Raw Moment :

Third Raw Moment :

Fourth Raw Moment :

(Ans)

4.3 RELATION BETWEEN CENTRAL MOMENTS AND

RAW MOMENTS

For grouped data, these results can be proved by replacing

4.4 SKEWNESS

Skewness refers to deviation from (or lack of) symmetry. A curve

which is not symmetric about any central vlaue on both the sides is called

skewed curve. When data is perfectly symmetrical about both the sides,

mean, median and mode co inicide at the central point. In case of

skewness, they change their position relative to each other.

Skewness can positive or negative.

Skewness measurement can be Absolute or Relative.

Absolute measures of Skewness :

There are two absolute measure s.

1)Karl Pearson’s measure of Skewness = Mean -Mode

2)Bowley’s measure of Skewness =

,munotes.in

## Page 36

36Where,

Relative measures of Skewness :

There are three relative measures of Skewness.

1)

2)

Bowley’s coefficien t of Skewness lies between -1 to +1

3)

Ex 2: Find Karl Pearson’s coefficient of Skewness for 4, 5, 3, 5, 5

Solution : Mean =

Mode = 5

(negavive skewness)

(Ans)

Ex 3: Find Bowley’s coefficient of Skewness fo r the following data.

Score 0-2020-4040-6060-8080-100

Number of student 15 25 32 35 16munotes.in

## Page 37

37Solution :

Score 0-2020-4040-6060-8080-100

Number of student 15 25 32 35 16

cf 15 40 72 107 123

To find Q1 :

Select cumulative frequency value highe r or equal to Rank,

To find Q2 :

Select cumulative frequency value higher or equal to Rank,

To find Q3 :

Select cumulative frequency value higher or equal to Rank,

,slight negative Skewness (Ans)munotes.in

## Page 38

38Ex 4 : Find Karl Pearson’s coefficient of Skewness

Range f

20-40 15

40-60 20

60-80 35

80-100 12

100-120 5

Solution :

Range FX fX

20-40 1530 450 900 13500

40-60 2050 1000

2500 50000

60-80 3570 2450

4900 171500

80-100 1290 1080

8100 97200

100-120 5110 550 12100 60500

Total 87 5530 28500 392700

The curve is slightly negatively skewed (Ans)

4.5 KURTOSIS

Normal distrbution curve is bell shaped in nature. But two

distribution may have symmetry, but their peakedness may vary. One may

have more height than the other. This characteristic is known as Kurtosis .

The main reason for this variation in peak is concentration of data around

the mean value. The curve will have higher peak for smaller standard

deviation.munotes.in

## Page 39

39

A distribution that is peaked in the same way as any normal

distribution is termed as Mesokurtic .

ALeptokurtic distribution is one with higher peak compared to

Mesokurtic distribution. The curne has higher p eak and is thin.

In contrast to Leptokurtic distribution, Platykurtic distribution is

flattened from top and has broad appearance compared to Mesokurtic

curves.

Measure of Kurtosis :

For Mesokurtic distribution,

,a n d

For Leptokurtic distribution,

,a n d

For Platykurtic distribution,

,a n d

Both

are unit free parameters and are independent of

change of scale and change of origin.

4.6 SUMMARY

1) Mom ents describe various parameters

2) Raw moments and Central moments can be related with various

formulas

3) Skewness represent extent of lack of symmetry in un symmetrical

distributions

4) Karl Pearson’s measure of Skewness and Bowley’s co efficient of

Skewness are measures of Skewness

5) Kurtosis represent thinness or flattened but symmetrical normal

distribution curves

6) Kurtosis can be Mesokurtic, Laptokurtic or Platykurticmunotes.in

## Page 40

404.7 EXERCISE

1) Expian Karl Pearson’s co -efficient of Skewness.

2) Find Karl Pearson’s coefficient of Skewness for 12, 14, 13, 16, 18

3) Find Bowley’s coefficient of Skewness for the following data.

Score 0-1010-2020-3030-4040-50

Number of student 23 42 45 40 12

4) Given

find

4.8 LIST OFREFERENC ES

1) Probability, Statistics, design of experiments and queuing theory with

applications of Compter Science, S. K. Trivedi, PHI

2) Applied Statistics, S C Gupta, S Chand

munotes.in

## Page 41

415

CORRELATION AND REGRESSION

ANALYSIS

Unit Structure

5.0 Objective

5.1 Int roduction

5.2 Correlation

5.2.1Scatter plot

5.2.2 Karl Pearson’s coefficient of Correlation

5.2.3 Properties of Correlation coefficient

5.2.4 Merits and Demerits of Correlation coefficient

5.2.5 Rank Correlation

5.3Regression

5.3.1 Linear Regression using method of least squares

5.3.2 Regression coefficient

5.3.3 Coefficient of determination

5.3.4 Properties of Regression coefficients

5.4 Summary

5.5 Exercise

5.6 List of References

5.0 OBJECTIVE

Correlation, as name suggests correlates two parame ters.

Statistically, Correlation coefficient gives an estimate of extent of

correlation between these two parameters (or quantities). One can

correlate score in final exam with the number of hours of study during the

term.

Regression is an estimation tech nique. It uses historical data to

estimate the possible value of that parameter in future. Regression analysis

helps to allocate resources based on estimation of the parameter like

estimation of future sales or estimation of future climatic condition.

5.1INTRODUCTION

Correlation can be measured statistically by Coefficient of

Correlation or even Scatter graph can be used.

Regression equation can be obtained either by method of least

squares or one can even use Regression coefficient.munotes.in

## Page 42

425.2 CORRELATION

Correlation analysis provides information about changes in one

parameter with reference to changes in othe rparameter. When one

variable increases, the other also increases (may be in different extent),

then the correlation is positive. In contrast to this, when variable increases,

the other dcreases, the correlation can be termed negative. There can

instances when there is no correlation between two parameters.

Correlation can be represented by :

1)Scatter Graph (Graphical representation) or

2)Karl Pearson’s co efficient of correlation (r) which is a stastical

measure of correlation

5.2.1 SCATTER GRAPH

Scatter Graph, also called X -Y plot gives following information about

two paratemers :

1)Shape (linear or non linear)

2)Extent of correlation

3)Nature of correlation l ike positive, negative or no correlation

Ex 1 : Plot Scatter Graph and comment.

XY

312

515

832

935

1245

Solution :

Comment : There seems to be high positive and linear relationship

between X and Y

(Ans)munotes.in

## Page 43

43Ex 2 : Plot Scatter Graph and comment.

XY

5612

4515

3232

2235

1245

Comment : There seems to be high negative and linear relationship

between X and Y

(Ans)

Ex 3 : Plot Scatter Graph and comment.

XY

512

1615

332

2235

145munotes.in

## Page 44

44Solution :

Comment : There seems to be slight negative or no correlation between X

and Y

(Ans)

Merits and Demerits of Scatter Graph

Merits :

1)Scatter Graph is easy to plot

2)It is also easy to understand and interpret general trend

3)Non linear relation can be easily detected

4)Scatter graph can very easily spot some abnormal values which are

ot consistent with rest of the values

Demerits :

1)Scatter graph does not give mathematical (or numerical) value of the

correlation, hence can not be used in further calculat ions, except for

visual observations

2)This method is useful for relatively small number of observations

3)It can not be applied to qualitative data whose numberical values are

not available like emotions, sentimets correlation can not be

represented by Scatte r Graph as no numerical values are available

5.2.2 KARL PEARSON’S COEFFICIENT OF CORRELATION

Karl Pearson’s coefficient of correlation (r) is used to find tpe of

correlation i.e. positive, negative or no correlation and also extent of

correlation like st rong, medium or weak correlation.

It is a numerical measure of correlation and is very useful in

statistical analysis.munotes.in

## Page 45

45Basic definition of ris

But, working formula for ris,

Ex 4 : Find Karl Pearson’s coefficient of correlation

XY

312

515

832

935

1245

Solution :

XY XY X² Y²

312 36 9 144

515 75 25 225

832 256 64 1024

935 315 81 1225

1245 540 144 2025

Total 37139 1222 323 4643

n = 5, number of ordered pairs

There is very strong posi tive correlation between X and Y (Ans)

Ex 5 : Find Karl Pearson’s coefficient of correlation

XY

5612

4515

3232

2235

1245munotes.in

## Page 46

46Solution :

XY XY X² Y²

5612 672 3136 144

4515 675 2025 225

3232 1024 1024 1024

2235 770 484 1225

1245540 144 2025

Total 37139 1222 323 4643

n = 5, number of ordered pairs

There is very strong negative correlation between X and Y (Ans)

Ex 6 : Find Karl Pearson’s coefficient of correlation

XY

512

1615

332

2235

145

Solution :

XY XY X² Y²

512 60 25 144

1615 240 256 225

332 96 9 1024

2235 770 484 1225

145 45 1 2025

Total 47139 1211 775 4643

n = 5, number of ordered pairs

There is slight negative correlation between X and Y (Ans)munotes.in

## Page 47

475.2.3 PROPERTIES OF KARL PEARSON’S COEFFICIENT OF

CORRELATION

1) Correlation coefficient lies between -1a n d+ 1

2) Correlation coefficient is independent of change of origin and scale

3) If variables are independent then they are uncorrelated (r nea rz e r o ) ,b u t

the converse is not true

4) Sometimes, correlation value may mislead, as there may be some value

of correlation by chance, but actually there is no evidence of

correlation

5.2.4MERITS AND DEMERITS OF COEFFICIENT OF

CORRELATION

Merits :

1)It iseasy to understand and easy to calculate

2)It indicates type of correlation i.e. negative, positive or no correlation

3)It also gives clear information about extent of correlation, +1 for

perfect positive and -1 for perfect negative correlation

Demerits :

1)Itcan mislead as higher correlation does not always mean close

relationship. Two variables can have high value of correlation but

may not actually have any relatinship

2)It is affected by extreme values of data set

3)Non linear relation is not very clearly indic ated by correlation

coefficient, whereas it is vlearly seen in Scatter plot

5.2.5 RANK CORRELATION

Rank correlation coefficient measures the degree of similarity

between two rankings.

For example, in a singing competition, two judges may give their

independent opinion about the participants through ranking, say 1, 2, 3 etc.

With the Rank correlation coefficient, one can find the extent to which

these two judges agree on the performance of the participant.

Spearman’s Rank Correlation

Where d is differ ence in Rankmunotes.in

## Page 48

48Ex 7 : Find Spearman’s Rank Correlation

R1 R2

1 2

2 3

3 1

4 5

5 4

Solution :

R1 R2 d=R 1 –R2

1 2 -1 1

2 3 -1 1

3 1 2 4

4 5 -1 1

5 4 1 1

Total 8

(Ans)

Spearman’s Rank Correlation when Ranks are repe ated

Where d is difference in Rank

5.3 REGRESSION

Regression is an estimation technique. It uses historical

data/information to estimate/predict near future value of that parameter.

For Example, score of a student in Mathematics exam can be predicted

based on student’s performance in a few previous years.munotes.in

## Page 49

49Regression line :

If X is independent variable, and Y is dependent variable, then the

Regression line can be given as :

Above Regression equation represents a strainght line. In practice ,

there can be non linear relationship between X and Y, in such a case, the

Regression equation can include square or cube or higher degree terms

also.

Regression Equation actually approximates and straightens the

point orientation by introducing some err or for alignment of the points to

get a straight line .i.e. Regression line.

5.3.1 LINEAR REGRESSION USING METHOD OF LEAST

SQUARES

Method of Least Squares is one of the methods to derive

Regression Equation.

Two parameters

can be

found out using two normal equations.

……….. Normal Equation I

……….. Normal Equation II

Solving these equations give values of aandbrequired to form

Regression Equationmunotes.in

## Page 50

50Ex 8 : Form Regression Equation for the following data set.

XY

512

1215

1532

2235

2545

Solution :

The two Normal equations are :

……….. Normal Equation I

……….. Normal Equation II

512 60 25

1215 180 144

1532 480 225

2235 770 484

2545 1125 625

Total 791392615 1503

Substituting these values in the two normal equations :

Solving simultaneously, or by method of substitution,

Substituting these values in the Regression Equation :

is the Regression Equation (Ans)

Ex 9 : Form Regression Equation for the following data set, and hence

estimate

XY

125

318

412

65

91munotes.in

## Page 51

51Solution :

The two Normal equations are :

……….. Normal Equation I

……….. Normal Equation II

12560 25

318180 144

412480 225

65770 484

911125 625

Total 2361166 143

Substituting these values in the two normal equations :

Solving simultaneously, or by method of substitution,

Substituting these values in the Regression Equation :

is the Re gression Equation

For

,

(Ans)

5.3.2 REGRESSION COEFFICIENT

Regression Coefficient b of Y on X

Regression Coefficient bof Y on X is given as :

Regression Equation can now be obtained as :

munotes.in

## Page 52

52Ex 10 : Find Regression Equation using Regression coefficient

XY

213

324

454

665

972

Solution :

21360 25

324180 144

454480 225

665770 484

9721125 625

Total 2361166 143

Regression Coefficient bof Y o n X is given as :

and

Regression Equation can now be obtained as :

is the Regression Equation (Ans)

Regression Coefficient b of Y on X

Regression Coefficient bof X on Y is given as :

Regress ion Equation can now be obtained as :

munotes.in

## Page 53

535.3.3 COEFFICIENT OF DETERMINATION

The Coefficient of detrmination,

,is a paramter used to judge

how well the estimated Regression line fits all the data, where

,is Karl

Pearson’s coef ficient of Correlation.

If the Regression line passes through all or most points, then

coefficient if determination will be close to 1.

Since,

Significance of coefficient of detrmination

1)It gives the strength of linear relaionship between two variables

2)It gives confidence to obtain variable to be predicted from the

indepndent variable

3)The coefficient of determination is the ratio of explained variation to

toal variation

4)It represents the quantum of data that is closest to the line of best f it

5)It is a measure of how well the Regression line represents the data

5.3.4 PROPERTIES OF REGRESSION COEFFICIENT

1)The point

lies on both the Regression lines

2)In case of perfect correlation between two variables,

or

3)Slope of Regression equation Y on X is given as,

wheras, slope

of Regression equation X on Y is given as

munotes.in

## Page 54

544)The angle between two Regression lines is given as,

5.4 SUMMARY

1)Correlation between two paramet ers can be represented either by

Scatter Graph or Karl Pearson’s coefficient of Correlation (r) can be

used

2)Karl Perason’s coefficient of correlation ranges between -1 to +1.

Negative correlation has negative value of r and positiove correlation

has positi ove value of r

3)Regression line helps to estimate or predict near future value of the

dependent parameter using historical values of the independent

variable

4)Regression line can be found out using method of least squares or

using Regression coefficient meth od

5)Coefficient of determination helps to understand how well is the

regression line fits or covers all or most data points

5.5 EXERCISE

1) Plot Scatter Graph and comment

X Y

201 34

226 45

230 56

312 53

340 62

357 64

2) Find Karl Pearson’s coeffi cient of correlation

XY

5512

4310

327

244

183

111munotes.in

## Page 55

553) Find Spearman’s Rank Correlation

R1 R2

1 4

2 3

3 2

4 1

5 5

4) Find Regression Equation for the following data set, using method of

least squares

XY

1212

1834

2667

3487

53106

66134

5) Find Regression Equation using Regression coefficient

XY

14

622

845

1077

1187

5.6LIST OFREFERENCES

1) Probability, Statistics, design of experiments and queuing theory with

applications of Compter Scienc e, S. K. Trivedi, PHI

2) Applied Statistics, S C Gupta, S Chand

munotes.in

## Page 56

566

PROBABILITY

Unit Structure

6.0 Objective

6.1 Introduction

6.2 Some basic definitions of Proabaility

6.3Permutations and Combinations

6.4Classical and axiomatic definitions of Pro bability

6.5 Addition Theorem

6.6Conditional Probability

6.7 Baye’s Theorem

6.8 Summary

6.9 Exercise

6.10 List of References

6.0 OBJECTIVE

The study of Probability helps learner to find solution to various

types problems which have some uncertainty in t heir occurence. Thie

shapter explains various definitions, concept and terms used in probabiluty

study in detail.

Learner should be able to understand and find solution to various

problems for which probability theory gives reasonably good solution.

6.1INTRODUCTION

Study of Probability is the study of chance. Probability theory is

widely applied to understand economic, social as well business problems.

Refer to the statements used by us in our daily life :

1)The train may get delayed

2)There is a chance of getting distinction in Mathematics by Mahesh

3)Asha may come on time today

Such statements are commonly used by all of us. One can

systematically study such probable events using principles of Probability

discussed in this chapter.munotes.in

## Page 57

576.2 SOME BASIC DEFINITION S OF PROBABILITY

Experiment :An experiment is an action that has more than one posiible

outputs

For Example :

1)Tossing a coin gives either a Head or a Tail

2)Throwing a die gives any one number from 1 to 6 on top face of the

die

3)A student appearing for an exam may pass or may fail exam

Experiment may be random or deterministic.

The output of the random experiment changes and occurs

randomly without any bias. In random experiments, all outcomes are

equally likely. For example, tossing a coin

The outcome o f the deterministic experiment does not change

when performed many times. For example, counting number of windows

of a particular room

Outcome :The result of an experient is called outcome. For example,

counting number of students in a class

Trial :Performing an experiment is called taking a trial

Sample Space :The collection of all possible outcomes is called sample

space of that experiment, For example, drawing a ball from a box having

three balls of Red, Blue and Green colours has a sample space of ba lls of

Red, Blue and Green colours. Sample space is demoted by letter S

Sample point :Each outcome of the sample space is called sample point.

The total number of sample points are denoted as n(S)

Finite sample space :When the number of outcomes are fini te, the

sample space in finite sample space. For example, number of students in

Statistics class of a college

Countably infinite sample space :When the number of elements in a

sample space are infinite, the sample space is said to countably infinite

samp le space. For example, set all all natural numbers

Exhaustive outcomes :Outcomes are exhaustive if they combine to be

the entire sample space.For example, outcomes Head and Tail are

exhaustive outcomes, when a coin is tossed

Event :Any subset of sample s pace associated with random experiment is

called an Event. Fro example, for a sample space={1, 2, 3, 4, 5}, an event

A can be “getting and odd number” and can be written as A={1, 3, 5}munotes.in

## Page 58

58Types of Event :Events can be described as given below :

1)Simple event : An event having only one outcome is called simple

event. For example, the evet of getting a head when a coin is tossed

2)Impossible event : The event corresponding to null set is called an

impossible event. For example, an event of getting a number more

than 6 when a die is thrawn

3)Sure event : The event corresponding to the sample space is called sure

event. For example, an event of getting either a head or a tail when a

fair coin is tossed

4)Mutually exclusive events : Two or more events are said be mutuall y

exclusive events if they do not have a sample point in common. For

example, an event of getting an even and another event of getting an

even number when a die is rolled

5)Exhaustive events : The events are said to be exhaustive events if

occurrence of any one event is surely going to take place. For example,

event of getting either red or black card when a card is drawn from a

pack of cards

6)Equally likely event : When all events have same chance of occurrence

then the events are equally likey. For example, getting a Head or a Tail

when an uniased coin is tossed, are called equally likely events

7)Independent events : Two or more events are said to be independent

events if one of them is not affected by occurrence of any other events.

i.e. P(A/B)=P(A)

6.3 PER MUTATIONS AND COMBINATIONS

Factorial :Factorial of a real number

is written as

such that

Ex 1 : Find

Solution :

(Ans)

Permutation :Permutation means arrangement of objects in different

ways. Fo r example, out of three objects A, B and C taken two at a time

can be arranged as AB, BA, BC, CB, CA, AC. We can arrange in six

different ways, as order or sequence of objects in Permutations is

important. So, if n objects are are arranged taken r at a tim e can be written

as,

Ex 2 : Find

Solution : take

(Ans)munotes.in

## Page 59

59Ex 3 : How many ways are there for eight men and five women to stand in

a row so that no two women stand next to each other.

Solution :

Eight men can be arranged in

ways.

Five women can be arranged in 9 ways as shown below :

*M * M * M * M * M * M * M * M *

Here * represents a place for a woman, and M represents a place for man.

Five women can be arranged in 9 places in

ways.

So, together eight men andfive women can be arranged such that no two

women stand together as :

Total number of ways =

ways

(Ans)

Ex 4 : In Hhw many ways can the letters of the word ‘MOUSE’ arranged,

where meaning/spelling does not ma tter.

Solution :

The words can be arranged in

ways. (Ans)

Combination :Combination is a selection of objects without consideting

the order of arrangements. For Example, for three objects A, B and C,

when two objects are taken at a time, the arrangement can be AB, BC and

AC. Order or sequence of arranements is not important in case of

Combination. So, Combination of n objects taken r at a time can be

written as,

Ex 5 : Find

Solution : take

(Ans)

Ex 6 : Find

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60Soluti on :

Also,

(Ans)

Ex 7 : In how many ways can a committee of 2 officers and 3 clerks can

be made from 4 officers and 10 clerks.

Solution : This can be done in

ways

ways (Ans)

6.4 CLASSICAL AND AXIOMATIC DEFINITIONS OF

PROBABILITY

Classical definition of Proability

When a random experiment is conducted having sample space S

having n(S) equally likely outcomes, the event A having n(A) favourable

outcomes , the probability of occurrence of event A is given as P(A) such

that :

Some inportant points regarding Probability definition are :

1)The sum of all probabilities in the sample space is 1 (one)

2)The probability of an impossible event is 0 (zero)

3)The probab ility of a sure event is 1 (one)

4)The probability of not occuring an event is 1 –probability of

occuring an event. i.e.

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61Ex 8 : Write down sample space for each of the following cases

1) A coin is thrawn three times

2) A coin is thrawn three times and number of heads in each thraw is

noted

3) A tetraheadron (a solid with four traingular surfaces) whose sides are

painted red, red, blue and green. The color of the side touching the

gound is noted

4) Blood group of husband and wife are tested and noted

Solution

1)

2)

3)

4)

Ex 9 : Thre eunbiased coins are tossed. What is the probability of getting

at least one Head.

Solution :

Sample Space,

Let A be the event of getting at least one Head

(Ans)

Ex 10 : Nine tickets are marked numbers 1 to 9. One ticket is drawn at

random. What is the probability that the number is an odd number.

Solution :

(Ans)munotes.in

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62Ex 11 : An urn contains 8 blue balls, 7 green balls and 5 red balls. One

ball is drawn at random, what is the probability that it is (a) a red ball, (b)

a blue ball.

Solution :

(a)Let A be the event of getting a red ball

(b)Let B be the event of getting a blue ball

Ex 12 : From a well shuffeled pack of cards, a card is drawn at random.

What is the probability that the drawn card is a red card

Solution :

Let A be the event of getting a red card

(Ans)

Ex 13 : What is the pro bability of getting a sum nine (9) when two dice

are thrawn

Solution :

Let A be the event of getting a sum nine (9)

(Ans)

Ex 14 : The Board of Directors of a company wants to form a quality

management committee to monitor quality of their products. The company

has 5 scientists, 4 engineers and 6 accountants. Find the probability that

the committee will have 2 scientists, 1 engineer and 2 accountants.

Solution :

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63Let A be the event of having 2 scientists, 1 engineer and 2 accountant s

(Ans)

Axiomatic definition of Proability

Suppose, for an experiment, S is the sample space containing outcomes,

, then assigning a real number

to each

uch that

1)

2)

6.5 ADDITION THEOREM

If A and B are two events defined on sample space, S then

a)Addition theorem can also be explained by Venn diagram

b)If two events are mutually exclusive, then

c)For three events,

Ex 15 : An integer is chosen at random from 1 to 100. Find the probability

that it is multiple of 5 or a perfect square

Solution :

Let A be the event of getting a number multiple of 5

Let B be the event of getting a perfect square

A

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64

By addition theorem,

Required p robability of getting a multiple of 5 or a getting a perfect square

is

(Ans)

Ex 16 : A card is drawn at random from a pack of cards. Find the

probability that the drawn card is a diamond or face card.

Solution :

Let A be the event of getting a diamond card

Let B be the event of getting a face card

By addition theorem,

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65Required probability of getting a multiple of 5 or a getting a perfect square

is

(Ans)

6.6 CONDITIONAL PRO BABILITY

Let there be two events A and B. The probability of event A given

that event B has occurred is known as conditional probabilityof A given

that B has occurred and is given as :

Ex 17: Given

.F i n d

Solution :

(Ans)

Ex 18 : Find the probability that a single toss of die will result in a number

less than 4 if it is given that the toss resulted in an odd number.

Solution : Let event A be the toss resulting in an odd number

And let event B be getting the number less than 4

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666.7 BAYE’S THEOREM

Let

be a set of mutually exclusive events that

together form the sample space S. Let

be any event from the same

sample space. Then Baye’s theorem st ates that

Ex 19 : In a toy factory, machines

manufacture

respectively 25%, 35% and 40% of total toys. Of these 5%, 4% and 2%

are defective toys. A toy is selected at random and is found to be

deefctive. What is the probability that it w as manufactured by machine

Solution :

Let

be any event that the drawn toy is defective.

We have to find

Required probability is 0.40 (Ans)munotes.in

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676.8 SUMMARY

1)

2)

3)

4)

5)

6)

6.9 EXERCISE

1) One card is drwan at random from a pack of cards. What is the

probability that it is a King or a Queen.

2) Find

3) Given an equiprobable sam ple space

,a n da n

event

Fnd

)

4) Given,

Find

5) A class has 40 boys and 20 girls. How many ways a class representative

(CR) be selected such that the CR is either a boy or a girl

6)From a set of 16 tickets numbered from 1 to 16, one ticket is drawn at

random. Find the probability that the number is divisible by 2 or 5

7) A car manufacturing company has two plants. Plant A manufactures

70% of the cars and the plant B manufactures 30 % of the cars. 80%

and 90% of the cars are of standard quality at plant A and plant B

respectuvely. A car is selected at random and is found to be of

standard quality. What is the probability that is was manufactured in

plant A

6.10 LIST OFREFERENCES

1)Probability, Statistics, design of experiments and queuing theory with

applications of Compter Science, S. K. Trivedi, PHI

2) Applied Statistics, S C Gupta, S Chand

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