Taxmann's Basics in Statistics & Quantitative Reasoning

P REFACE TO SECOND EDITION The primary objective of this text book is to equip the students with various mathematical techniques and application of these techniques in business problems.

The emphasis is made on the concept and application rather than deriving the various formulae. The book aims at laying a foundation for developing their skills in analyzing and interpreting mathematical statements. The questions given in the book are multi-objective which help the students appear in various entrance examinations.

The basic concepts are explained to the students with a variety of illustrations and hints. Thus the approach is to communicate the practical aspects of MATHEMATICS and STATISTICS. We, gratefully acknowledge the inspiration, encouragement and valuable suggestions received from the students and teachers. Finally, we express our sincere thanks to all those individuals who have been a source of inspiration and support personally and professionally. We thank Mr. Yadav of Taxmann Publication Pvt. Ltd., who has taken keen interest in the book and tried his level best to bring the book in time.

Lastly, we must concede that this book would never have been written without the support, encouragement and prodding of our family members. Many thanks to them.

We have tried our best to make this book free of errors, omissions and discrepancies. Still if you find any potential error or wish to seek any

I-7

PREFACE TO SECOND EDITION

I-8

clarifications, please mail us at the E-mail address given below. Your suggestions will be highly appreciated and duly incorporated in the subsequent edition.

DR. S.R. ARORA

srarora@yahoo.com

DR. KAVITA GUPTA

gupta_kavita31@yahoo.com

CHAPTER

HEADS PAGE

About the Authors

I-5

Preface to Second Edition

I-7

Contents

I-11

CHAPTER 1

: PERCENTAGE, PROFIT AND LOSS, RATIO AND PROPORTION

1

CHAPTER 2

: SAMPLING AND SAMPLING DISTRIBUTIONS

20

CHAPTER 3

: FORMATION OF FREQUENCY DISTRIBUTION AND THEIR GRAPHICAL REPRESENTATION

41

CHAPTER 4

: MEASURES OF CENTRAL TENDENCY

62

CHAPTER 5

: MEASURES OF VARIATION AND SKEWNESS

113

CHAPTER 6

: MEASUREMENT OF SCALE

198

CHAPTER 7

: SET THEORY

206

CHAPTER 8

: RELATIONS AND FUNCTIONS

220

CHAPTER 9

: THEORY OF PROBABILITY

238

CHAPTER 10 : PERMUTATION AND COMBINATIONS

270

CHAPTER 11 : PROGRESSIONS AND SERIES

292

CHAPTER 12 : INDEX NUMBERS

311

CHAPTER 13 : MATHEMATICS OF FINANCE

329

CHAPTER 14 : APPLICATION OF DERIVATIVES

368

I-9

CHAPTER HEADS

I-10 PAGE

TABLES TABLE I

: AMOUNT OF AN ANNUITY

417

TABLE II

: PRESENT VALUE OF AN ANNUITY

425

TABLE III

: TABLE OF e AND e

433

TABLE IV

: LOGARITHMS

438

x

-x

CONTENTS PAGE

About the Authors Preface to Second Edition Chapter-heads

I-5 I-7 I-9

CHAPTER 1 PERCENTAGE, PROFIT AND LOSS, RATIO AND PROPORTION 1.1

Introduction

1

1.2

Percentage

1

1.3

Profit and Loss

4

Objective Type Questions

7

Fill in the Blanks

10

True/False

12

Answers to Objective Type Questions

13

Answers to Fill in the Blanks

18

Answers to True/False

19

CHAPTER 2 SAMPLING AND SAMPLING DISTRIBUTIONS 2.1

Introduction

20

2.2

Descriptive and Inferential Statistics

20

2.3

Population and Sample

21

2.4

Census versus Sample Method

22

2.5

Statistics and Parameters

22

I-11

CONTENTS

I-12 PAGE

2.6 Sampling Methods 2.7 Simple Random Sampling 2.8 Systematic Sampling 2.9 Stratified random sampling 2.10 Cluster Sampling 2.11 Multi-stage sampling 2.12 Judgment Sampling 2.13 Convenience Sampling 2.14 Quota Sampling 2.15 Bias and Error in Sampling Objective Type Questions Fill in the Blanks True/False Answers to Objective Type Questions Answers to Fill in the Blanks Answers to True/False

24 25 27 28 29 29 30 31 31 32 34 37 38 39 40 40

CHAPTER 3 FORMATION OF FREQUENCY DISTRIBUTION AND THEIR GRAPHICAL REPRESENTATION 3.1 Introduction 3.2 Frequency Distribution 3.3 Tabulation of Data 3.4 Graphical Representation of a Frequency Distribution Objective Type Questions Fill in the Blanks True/False Answers to Objective Type Questions Answers to Fill in the Blanks Answers to True/False

41 41 45 46 55 57 59 59 60 61

CHAPTER 4 MEASURES OF CENTRAL TENDENCY 4.1

Introduction

62

4.2

Definition of Average

62

4.3 Properties of a Good Average 4.4 Various Measures of Central Tendency Objective Type Questions

63 63 95

I-13

CONTENTS PAGE

Fill in the Blanks True/False Answers to Objective Type Questions Answers to Fill in the Blanks Answers to True/False

101 102 103 111 112

CHAPTER 5 MEASURES OF VARIATION AND SKEWNESS 5.1

Introduction

113

5.2

Meaning of Variation or Dispersion

113

5.3

Characteristics of a Good Measure of Variation

114

5.4

Types of measures of variation

114

5.5

Quartiles, Deciles and Percentiles

115

5.6

Measures of Variation

122

5.7

Coefficient of Variation

144

5.8

Meaning of Skewness

153

5.9

Types of Skewness

154

5.10

Difference between variation and skewness

155

5.11

Measures of Skewness

155

5.12

Karl Pearson’s Coefficient of Skewness

156

5.13

Bowley’s Coefficient of Skewness

167

5.14

Kurtosis Objective Type Questions Fill in the Blanks True/False Answers to Objective Type Questions Answers to Fill in the Blanks Answers to True/False

172 173 182 184 185 195 196

CHAPTER 6 MEASUREMENT OF SCALE 6.1

Introduction

198

6.2

Levels of measurement

198

6.3

Different Scales of Measurements Objective Type Questions Answers to Objective Type Questions

198 201 204

CONTENTS

I-14 PAGE

CHAPTER 7 SET THEORY 7.1 Introduction 7.2 Meaning of a set 7.3 Representation of Sets 7.4 Types of Sets 7.5 Operations on Sets 7.6 Applications of Set Theory 7.7 Cartesian product of sets Objective Type Questions Fill in the Blanks True/False Answers to Objective Type Questions Answers to Fill in the Blanks Answers to True/False

206 206 207 207 210 212 212 215 217 217 217 219 219

CHAPTER 8 RELATIONS AND FUNCTIONS 8.1 Introduction 8.2 Relation 8.3 Types of Relations 8.4 Functions 8.5 Algebra of Functions 8.6 Types of Functions 8.7 Functions Related to Business and Economics Objective Type Questions Fill in the Blanks Answers to Objective Type Questions Answers to Fill in the Blanks

220 221 221 222 222 223 225 230 232 233 237

CHAPTER 9 THEORY OF PROBABILITY 9.1

Introduction

238

9.2

Meaning of Probability

239

9.3

Basic Terminology

239

9.4

Different Approaches to Probability

240

I-15

CONTENTS PAGE

9.5 Addition Theorem of Probability 9.6 Multiplication Theorem of Probability 9.7 Conditional Probability 9.8 Bayes’ Theorem Objective Type Questions Fill in the Blanks True/False Answers to Objective Type Questions Answers to Fill in the Blanks Answers to True/False

241 242 243 243 253 258 260 261 267 269

CHAPTER 10 PERMUTATION AND COMBINATIONS 10.1

Introduction

270

10.2

Factorial

270

10.3

Fundamental Principle of Multiplication

271

10.4

Fundamental Principle of Addition

271

10.5

Permutations

272

10.6

Permutations Under Different Conditions

273

10.7

Combinations

275

Objective Type Questions

277

Fill in the Blanks

281

True/False

283

Answers to Objective Type Questions

284

Answers to Fill in the Blanks

289

Answers to True/False

290

CHAPTER 11 PROGRESSIONS AND SERIES 11.1

Introduction

292

11.2

Sequence and Series

292

11.3

Progressions

292

11.4

Arithmetic Progression (A.P)

293

11.5

Geometric Progression

301

11.6

Harmonic Progression

308

CONTENTS

CHAPTER 12

I-16 PAGE

INDEX NUMBERS 12.1 12.2 12.3 12.4 12.5 12.6

Introduction Meaning of Index Numbers Uses of Index numbers Methods of constructing index numbers Weighted Aggregative Index Value Index

311 311 311 312 317 324

CHAPTER 13 MATHEMATICS OF FINANCE 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11

Introduction Basic Terminology of Finance Simple Interest Compound Interest Interest Compounded Continuously Compound Amount at Changing Rates Present Value or Capital Value Annuity Amount or Future Value of an Ordinary Annuity Present Value of an Ordinary Annuity Amortization of Loans

329 329 330 334 338 340 345 347 348 353 362

CHAPTER 14 APPLICATION OF DERIVATIVES 14.1 14.2 14.3 14.4 14.5 14.6 Table I

Introduction Average cost and marginal cost Average revenue and marginal revenue Marginal revenue product Marginal propensity to consume Applied max-min problems TABLES : Amount of an annuity

368 368 375 381 382 385 417

Table II : Present value of an annuity

425

Table III : Table of e and e

433

Table IV : Logarithms

438

x

-x

4

C H A P T E R

MEASURES OF CENTRAL TENDENCY

LEARNING OBJECTIVES By the end of this chapter, you will be able to understand:

Meaning of average or central tendency

Properties of a good average

Measures of Central Tendency: Arithmetic mean, Median, Mode

Properties of different types of averages.

Determination of median and mode for grouped data graphically.

4.1 INTRODUCTION One of the objectives of statistical analysis is to determine various numerical measures which describes the inherent characteristics of a frequency distribution. The first of such measures is average. The term “average” is very commonly used in day to day conversation. The first and foremost objective of statistical analysis is to get one single value that represent or describes the entire data. Such a single value is called average or central value.

4.2 DEFINITION OF AVERAGE Different statisticians gave different definitions of average from time to time.

Some of them are: “Average is an attempt to find one single figure to describe whole of figures”.

-Clark “Averages are statistical constants which enable us to comprehend in a single effort the significance of the whole.”

-A.L. Bowley 62

63

VARIOUS MEASURES OF CENTRAL TENDENCY

Para 4.4

“An average is a single value within the range of the data that is used to represent all the values in the series. Since an average is somewhere within the range of the data, it is sometimes called a measure of central value”. - Croxton and Cowden

4.3 PROPERTIES OF A GOOD AVERAGE A good measure of central tendency should possess the following properties: 1. It should be rigidly defined. It means that the definition should be clear so that it leads to one and only one interpretation.

2. It should be easy to understand and simple to calculate. It should be so easy that even a non-mathematical person can calculate it.

3. It should be based on all the observations. It means that entire set of data should be used in computing average and there should not be any loss of information resulting from not using the available data.

4. It should be capable of further algebraic treatment. Average should be capable of further mathematical and statistical computations to expand or enhance its utility.

5. It should not be unduly affected by extreme observations. Average should be such that it should not be affected by the presence of one or two very small or very large observations.

6. It should not be affected too much by fluctuations of sampling. It should have sampling stability. By sampling stability, we mean that if we take different samples of same size from a large population and compute the average of each sample, we expect to get the same answer approximately.

There can be slight fluctuations in values of different samples.

4.4 VARIOUS MEASURES OF CENTRAL TENDENCY In this section, we will discuss the following measures of central tendency which are most commonly used in practice. (i) Arithmetic Mean – Simple and Weighted Mean (ii) Median (iii) Mode 4.4.1 Arithmetic Mean: The most popularly used measure of central tendency is arithmetic mean or simply mean. Arithmetic mean is of two types : 1. Simple arithmetic mean 2. Weighted arithmetic mean.

Para 4.4

MEASURES OF CENTRAL TENDENCY

64

Simple Arithmetic Mean or Mean A. In case of ungrouped data: 1. In case of individual observations: Let X1,...,Xn be the given observations.

Then arithmetic mean or A.M of these observations is denoted by X and is given by

X =

∑X

n where n is the number of observations.

Short-cut method:

X =A+

∑ d , where d

n where A is the assumed mean.

= X − A

2. In case of discrete frequency distribution: (a) Direct Method: The formula for calculating mean in a discrete series is : X =

∑ fX ∑f

where f = frequency , X denotes given observations. (b) Short-cut Method: X =A+

∑ fd , where d =X - A ∑f

where A is the assumed mean.

B. In case of grouped frequency distribution or continuous series: (a) Direct Method: X =

∑ fm ∑f

where m is the mid-value of each class interval and is given by m=

lower limit + upper limit 2

(b) Short-cut method: X =A+

∑ fd , where d =m - A ∑f

where A is the assumed mean , m is the mid-value of each class interval.

65

VARIOUS MEASURES OF CENTRAL TENDENCY

Para 4.4

(c) Step-deviation method or coding method: X =A+

∑ fu × i , where u = m − A i ∑f

where A = assumed mean, m = mid-value of each class interval, i = step factor

Properties of Arithmetic mean The following are a few important properties of arithmetic mean: 1. The sum of the deviations of the items from the arithmetic mean is always zero, i.e.,

∑ (X − X ) = 0

2. The sum of the squared deviations of the items from the arithmetic mean is minimum, i.e.,

∑ (X − X )

2

is minimum.

3. If each item of a series is increased or decreased by a constant k, then the arithmetic mean of the new series also get increased or decreased by k, i.e., New mean = X + k 4. If each item of a series is multiplied by a constant k then the arithmetic mean of the new series also gets multiplied by k, i.e., New mean = kX 5. Combined Arithmetic Mean : Consider two related groups such that N1 and N2 are the number of observations in first and second groups respectively.

Let X 1 and X 2 be their respective means. Then the mean of the two groups taken together or their combined mean X 12 is given by X 12 =

N 1X 1 + N 2 X 2 N1 + N 2

The formula can be extended to more number of groups.

Merits of Arithmetic Mean 1. It is simple to calculate and easy to understand.

2. It is based on each and every observation of the series.

3. It does not fluctuate with sampling.

4. It does not depend upon the position in the series.

5. It is capable of further algebraic treatment.

6. It is rigidly defined. Everyone will get the same answer when apply the formula of average.

Para 4.4

66

MEASURES OF CENTRAL TENDENCY

Demerits of Arithmetic Mean 1. It is unduly affected by extreme values, i.e., by the presence of very large and very small items. For instance, mean of 55, 54, 49, 50, 5 is 42.6 but 42.6 is not a single value that represent the whole of data as one single item 5 has affected the average so much.

2. It cannot be determined by inspection like mode and it cannot be located graphically.

3. In case of open-end classes where the lower limit of the first class interval and upper limit of the last class interval is not known, mean sometimes introduces error. In such cases, assumptions should be made regarding the size of the class interval of open-end classes. In such cases, median and mode are the most suitable averages.

4. Mean is not a suitable average in case of qualitative data such as honesty, beauty, voice quality etc. In such cases, rank correlation is computed.

5. Mean is not a good measure of central tendency in case of normal distribution and in case of U shaped distribution.

Example 1: (Mean in individual observations)You are given the marks obtained

by eight students of B.Com. Find the mean marks.

Roll No.

1

2

3

4

5

6

7

8

Marks:

56

60

62

53

78

45

51

61

Solution: The given data is in the form of individual observations.

X =

∑X n

56 + 60 + 62 + 53 + 78 + 45 + 51 + 61 8 466 = = 58.25 marks 8

=

Example 2: (Mean by shortcut method in individual observations): Compute

mean from the following data:

260.21, 260.22, 260.16, 260.17, 260.15, 260.17, 260.12, 260.15, 260.13, 260.12 Solution: Computation of Mean

X

d = X – A = X - 260

260.21

0.21

260.22

0.22

260.16

0.16

260.17

0.17

260.15

0.15

260.17

0.17

67

Para 4.4

VARIOUS MEASURES OF CENTRAL TENDENCY

X

d = X – A = X - 260

260.12

0.12

260.15

0.15

260.13

0.13

260.12

0.12

∑ d = 1.60 By short-cut method, X = A +

∑ d , where d =X - A

n 1.6 = 260 + = 260.16 10 Example 3: (Mean in discrete series by direct method) Compute the mean marks from the following data: Marks:

65

55

45

35

25

15

5

No. of students:

4

6

10

20

10

6

4

Solution:

Computation of Mean Marks(X)

No. of students (f)

fX

65

4

260

55

6

330

45

10

450

35

20

700

25

10

250

15

6

90

5

4

20

∑ f = 60

∑ fX = 2100

By direct method, X =

∑ fX ∑f

2100 = 35 marks 60 Example 4: (Mean in discrete series by short-cut method) Compute the mean from the following data: =

X:

9.5

8.5

7.5

6.5

5.5

4.5

3.5

f:

8

32

85

60

22

7

3

Para 4.4

68

MEASURES OF CENTRAL TENDENCY

Solution:

Computation of Mean by Short-cut method X

f

d= X – A =X – 6.5

fd

9.5

8

3

24

8.5

32

2

64

7.5

85

1

85

6.5

60

0

0

5.5

22

–1

– 22

4.5

7

–2

– 14

3.5

3

–3

–9

∑ f = 217

∑ fd = 128

By short-cut method, X =A+

∑ fd , ∑f

d =X - A

128 217 = 6.5 + 0.589 = 7.089 = 6.5 +

Example 5: (Mean in continuous series by direct method): Compute the mean

from the data given below:

Solution:

X

f

0-10 10-20 20-30 30-40 40-50

4 6 10 20 10

Computation of Mean by direct method X

f

Mid value(m)

fm

0-10

4

5

20

10-20

6

15

90

20-30

10

25

250

30-40

20

35

700

40-50

10

45

450

∑ f = 50

∑ fm = 1510

69

VARIOUS MEASURES OF CENTRAL TENDENCY

Para 4.4

By direct method, X =

∑ fm ∑f

1510 = 30.2 50 Example 6: (Mean in continuous series by step deviation method) Compute the mean wages of the workers from the data given below: =

Wages (in `)

No. of workers

40-60 60-80 80-100 100-120 120-140 140-160 160-180 180-200

10 15 28 32 20 10 8 5

Solution :

Computation of Mean wages Wages (X)

No. of workers(f)

Mid-value(m)

40-60 60-80 80-100 100-120 120-140 140-160 160-180 180-200

10 15 28 32 20 10 8 5

50 70 90 110 130 150 170 190

∑ f =128 By step deviation method, X =A+

∑ fu × i , where u= m − A = m − 110 i 20 ∑f

m −A i m − 110 = 20

u=

-3 -2 -1 0 1 2 3 4

fu

-30 -30 -28 0 20 20 24 20

∑ fu = -4

Para 4.4 = 110 − =

70

MEASURES OF CENTRAL TENDENCY

4 × 20 128

1510 50

Therefore, mean wages = ` 109.375 Example 7: (Mean in continuous series) The following table gives the income of

employees of a company. Compute the mean income.

Income (in thousand rupees)

No. of employees

Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70

5 22 48 60 83 100

Solution:

Computation of Mean income Income

Class No. of intervals(X) employees (f)

Midvalue(m)

u= =

m −A i

fu

m − 45 10

Less than 20 10-20

5

15

-3

-15

Less than 30 20-30

22-5 = 17

25

-2

-34

Less than 40 30-40

48-22 = 26

35

-1

-26

Less than 50 40-50

60-48 = 12

45

0

0

Less than 60 50-60

83-60 = 23

55

1

23

Less than 70 60-70

100-83 = 17

65

2

34

∑ f =100 By step deviation method, X =A+

∑ fu × i , where u= m − A = m − 45 i 10 ∑f

= 45 −

18 × 10 100

= 43.2 Therefore, mean income = ` 43.2 thousand

∑ fu

-18

Basics in Statistics & Quantitative Reasoning AUTHOR PUBLISHER DATE OF PUBLICATION EDITION ISBN NO NO. OF PAGES BINDING TYPE

: : : : : : :

S.R. Arora, Kavita Gupta TAXMANN July 2022 2nd Edition Reprint 9789356222960 460 PAPERBACK

Rs. : 500 | USD : 37

Description This is an authentic and comprehensive textbook covering the entire syllabus. This book equips the students with various mathematical techniques and the application of these techniques to business problems.

The emphasis is made on the concept and application rather than deriving the various formulae. The book lays a foundation for developing their skills in analyzing and interpreting mathematical statements.

This book aims to fulfil the requirement of students for undergraduate courses in commerce and management.

The Present Publication is the 2nd Reprint Edition, authored by Dr S.R. Arora & Dr Kavita Gupta, with the following noteworthy features:

·

[Simple, Systematic & Lucid Language] The subject matter is presented in a simple, systematic method and lucid language to explain the concept and application.

·

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·

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· ·

[Solved Examples] are included in each chapter

·

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the topics thorough understanding [Excercises with Hints & Detailed Answers] to understand the problem thoroughly the following factors:

o

Interaction of the author/teacher with their students in the

o

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o

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