Table Of ContentS
tat i s t i c a l
M
e t h o d s
(U.P. UNIFIED)
(for Ist year B.A./B.Sc. Statistics Students of All Colleges affiliated to
Universities in Uttar Pradesh)
By
Dr. Arun Kumar Dr. Alka Chaudhary
Reader & Head Reader
Dep't. of Statistics Dep't. of Statistics
M.M. (P.G.) College, Modinagar (U.P.) Meerut College, Meerut (U.P.)
KRISHNA Prakashan Media (P) Ltd.
KRISHNA HOUSE, 11, Shivaji Road, Meerut-250 001 (U.P.), India
Jai Shri Radhey Shyam
Dedicated
to
Lord
Krishna
Authors & Publishers
A B
bout the ook
Statistical Methods is an up-to-date reference book for B.Sc. (Statistics) Ist year Students
of Uttar Pradesh.
The key features are:
1. Covers 100% syllabus of B.Sc. (U.P. Unified).
2. A large number of solved problems have been listed chapter-wise for better and
easier understanding.
3. Numerous tables and diagrams are presented to rouse the interest of students
and also to present the content effectively.
4. The book aims at:
(i) Accuracy
(ii) Clarity
(iii) Simplicity.
(iv)
P
reface
This book on Statistical Methods has been specially written to meet the
requirements of unified syllabus of B.A. and B.Sc. Part-I students of all the
universities of Uttar Pradesh. The subject matter has been discussed in such
a simple way that the students will find no difficulty to understand it. The
proofs of various theorems and examples have been given with minute details.
Each chapter of this book contains complete theory and a fairly large number
of solved examples. Sufficient problems have also been selected from Various
University Examination Paper.
To improve the scope of the book Objective type questions viz. Multiple
choice questions, True/False and Fill in the blanks have also been added.
As usual, recent question paper of other universities have also been included
in the last with hints and solutions.
We have tried our best to keep the book free from misprints. The authors shall
be grateful to the readers who point out errors and omissions which, inspite of
all care, might have been there.
The authors will feel amply rewarded if the book serves the purpose for which
it is meant. Suggestions for the improvement of the book are always welcome.
Dr. Arun Kumar Dr. (Mrs.) Alka Chaudhary
49, Sona Enclave, R-17, Medical College Campus,
Modinagar (GZB) Meerut
(v)
A
cknowledgement
We are grateful to the participants of a numerous execution development
programmes for providing the stimulus to write this book. We acknowledge
our indebtedness to a large number of books and articles which we have
consulted during the compilation of this book.
Special thanks to Mr. Satyendra Kumar Rastogi (Managing Director)
Krishna Prakashan Media (P) Ltd., for this support. We also thank
Mr. Sugam Rastogi (Executive Director), for an excellent job in editing the
manuscript and entire K.P.M. team for publishing and bringing out the book in
this shape.
Last but not the least, we deeply express our thanks to the readers. Suggestions and
criticism would be highly appreciated and acknowledged.
— Authors
(vi)
Syllabus
Statistical Methods
U.P. UNIFIED (w.e.f. 2011-2012)
B.Sc Ist Year
Paper-III
Unit-I
Concept of Statistical population, Attributes and variables (Discrete and
Continuous). Different types of scales - nominal, ordinal, ratio and interval. Primary
data - designing a questionnaire and schedule, collection of primary data, checking
their consistency. Secondary data. Scrutiny of data for internal consistency and
detection of errors of recording. Presentation of data: classification, tabulation,
diagrammatic & graphical representation of grouped data. Frequency distributions,
cumulative frequency distributions and their graphical representations, histogram,
frequency polygon and ogives. Stem and Leaf plot. Box Plot.
Unit-II
Measure of central tendency and dispersion, merits and demerits of these measures.
Moments and factorial moments. Shephard's correction for moments. Skewness and
Kurtosis and their Measures. Measures based on quartiles. Bivariate data, Method of
least squares for curve fitting.
Unit-III
Correlation and Regression, Rank Correlation (Spearman's and Kendall's measure),
Intra-class correlation, correlation ratio. Partial and Multiple Correlation & Multiple
Regression for Trivariate data.
Unit-IV
Attributes-Notation and terminology, contingency table, class frequencies, and
ultimate class frequencies, consistency. Association of attributes, Independence,
Measure of association for 2×2 table. Chi-square, Karl Pearson's and Tschuprow's
coefficient of association. Contingency tables with ordered categories.
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B C
rief ontents
Dedication ....................................................................................................... (iii)
About the Book .................................................................................................. (iv)
Preface .............................................................................................................. (v)
Acknowledgement .............................................................................................. (vi)
Syllabus (B.Sc. Statistics - U.P. Unified) ............................................................ (vii)
Brief Contents ................................................................................................. (viii)
Detailed Contents ......................................................................................... (ix-xii)
Unit-I
Chapters ........................................................................................................ Pages
1. Definition, Functions, Limitations and Importance of Statistics .................... 01-14
2. Types of Data and Scales .................................................................................. 15-20
3. Collection and Scrutiny of Data ...................................................................... 21-34
4. Organisation of Data ....................................................................................... 35-58
5. Diagrammatic Representation of Data ............................................................ 59-74
6. Graphic Representation of Data ...................................................................... 75-90
Unit-II
7. Measures of Central Tendency ....................................................................... 91-162
8. Measures of Dispersion Skewness and Kurtosis .......................................... 163-214
9. Method of Least Squares and Curve Fitting ................................................ 215-236
Unit-III
10. Correlation and Regression ......................................................................... 237-294
11. Multiple and Partial Correlation ................................................................. 295-316
Unit-IV
12. Theory of Attributes .................................................................................... 317-352
Examination Paper ........................................................................................... 01-04
(viii)
D C
etailed ontents
Unit-I
Chapters ........................................................................................................................ Pages
1. Definition, Functions, Limitations and Importance of Statistics ............................ 1-14
1.1 Introduction 1
1.2 Definition of Statistics 1
1.2.1 Statistics as a Plural Noun 2
1.2.2 Statistics as a Singular Noun 3
1.3 Other Popular Definitions of Statistics 3
1.4 Function of Statistics 4
1.5 Limitation of Statistics 4
1.6 Distrust of Statistics 5
1.6.1 Reasons for Distrust in Statistics 5
1.7 Importance of Statistics 6
1.8 Statistical Tools Used in Economic Analysis 7
1.8.1 Stages in Statistical Investigation 7
Exercise 11
Answers 14
2. Types of Data and Scales ..................................................................................... 15-20
2.1 Introduction 15
2.2 Census and Sampling 15
2.2.1 Population and Sample 15
2.2.2 Census Enquiry and Sample Enquiry 16
2.3 Types of Data 16
2.3.1 Nominal Scale 16
2.3.2 Ordinal Scale 16
2.3.3 Interval Scale 16
2.3.4 Ratio Scale 16
2.3.5 Nominal and Ordinal Data 17
2.3.6 Time Series Data 17
2.3.7 Geographical Data 17
2.3.8 Discrete Data 17
2.3.9 Continuous Data 17
2.3.10 Frequency Data 17
2.3.11 Non Frequency Data 18
Exercise 18
Answers 20
3. Collection and Scrutiny of Data .......................................................................... 21-34
3.1 Introduction 21
3.2 Primary and Secondary Data 22
3.3 Method of Collection 22
3.3.1 Collection of Primary Data 22
3.3.2 Questionnaire and Schedule Making 25
3.3.3 Collection of Secondary Data 27
3.4 Scrutiny of the Data 28
3.4.1 Control Measures
Exercise 31
Answers 34
(ix)
4. Organisation of Data ........................................................................................... 35-58
4.1 Introduction 35
4.2 Classification 35
4.3 Object of Classification 36
4.4 Basis of Classification 36
4.4.1 Classification of Attributes 36
4.4.2 Classification of Variables (Quantitative Classification) 37
4.5 Frequency Distribution 38
4.6 Method of Construction of Discrete Frequency Distribution 38
4.7 Method of Construction of Continuous Frequency Distribution 39
4.8 Basic Principles for Forming Grouped Frequency Distribution 40
4.9 Sturges Rule for Number of Classes and Size of Class Interval 40
4.10 Cumulative Frequency Distribution 43
4.11 Tabulation 45
4.12 Types of Tables 47
4.13 Difference between Classification and Tabulation 49
Exercise 50
Answers 58
5. Diagrammatic Representation of Data ................................................................ 59-74
5.1 Introduction 59
5.2 Importance and Utility of Diagrams 59
5.3 Limitations of Diagrams 60
5.4 Rules for Constructing Diagrams 60
5.5 Types of Diagrams 60
5.5.1 One Dimensional Diagrams 60
5.5.2 Two Dimensional Diagrams or Area Diagrams 64
5.5.3 Pictograms or Pictorial Diagrams 67
5.5.4 Cartograms or Maps 67
5.6 Limitations of Diagrammatic Representation 68
Exercise 68
Answers 74
6. Graphic Representation of Data .......................................................................... 75-90
6.1 Introduction 75
6.2 Graphs of Frequency Distribution 75
6.2.1 The Histogram 75
6.2.2 The Frequency Polygon 79
6.2.3 The Frequency Curve 79
6.2.4 Cumulative Frequency Curve or Ogive 82
6.3 Stem and Leaf Diagram 85
6.4 Box Plot or Box and Whisker Diagram 86
Exercise 86
Answers 90
Unit-II
7. Measures of Central Tendency ............................................................ 91-162
7.1 Introduction 91
7.2 Objectives of Average 91
7.3 Characteristics of a Good Average 92
7.4 Various Measures of Central Tendency 92
7.4.1 Arithmetic Mean 94
7.4.2 Geometric Mean 105
7.4.3 Harmonic Mean 109
7.4.4 Median114
7.4.5 Graphical Determination of Median 118
7.5 Partition Values or Quantiles 119
7.5.1 Graphical Determination of the Partition Values 121
(x)
7.6 Mode 122
7.6.1 Calculation of Mode 123
Exercise 133
Answers 160
8. Measures of Dispersion, Skewness and Kurtosis .............................................. 163-214
8.1 Introduction 163
8.2 Objects and Importance of Dispersion 164
8.3 Characteristics for Satisfactory Measures of Dispersion 164
8.4 Absolute and Relative Measure of Variation 164
8.5 Measures of Dispersion 165
8.5.1 Range 165
8.5.2 Interquartile Range and Quartile Deviation 165
8.5.3 Mean Deviation or Average Deviation 167
8.5.4 Standard Deviation and Root Mean Square Deviation 171
8.6 Root Mean Square Deviation 172
8.7 Relation between s and S 173
8.8 Effect of Change of Origin and Scale on Standard Deviation 174
8.9 Combined Standard Deviation 183
8.10 Mathematical Properties of Standard Deviation187
8.11 Moments 187
8.12 Relation between mr and mr'188
8.13 Effect of Change of Origin and Scale on Moments 189
8.14 Sheppard's Correction on Moments 190
8.15 Charlier's Check 191
8.16 Pearson's b Coefficients and Fisher's g Coefficients 191
8.17 Skewness 196
8.18 Measures of Skewness 197
8.18.1 Measures of Skewness Based on Moments 197
8.18.2 Karl Pearson’s Coefficient of Skewness 198
8.18.3 Bowley’s Coefficient of Skewness 198
8.18.4 Kelly’s Coefficient of Skewness 199
8.19 Kurtosis 199
8.19.1 Measures of Kurtosis 199
Exercise 203
Answers 213
9. Method of Least Squares and Curve Fitting ..................................................... 215-236
9.1 Introduction 215
9.2 Method of Least Squares 215
9.3 System of Linear Equations 216
9.4 Curve Fitting 218
9.4.1 Fitting of a Straight Line 219
9.4.2 Fitting a Parabola of Second Degree 219
9.4.3 Change of Origin and Scale 221
9.4.4 Fitting of Exponential Curves 226
9.4.5 Fitting of Logarithmic Curve 227
9.4.6 Advantages and Disadvantages of Least Squares Method 231
Exercise 232
Answers 236
Unit-III
10. Correlation and Regression ............................................................................. 237-294
10.1 Univariate and Bivariate Distributions 237
10.2 Correlation 237
10.3 Types of Correlation 238
10.3.1 Positive or Negative Correlation 238
10.3.2 Linear and Non-Linear Correlation 239
10.4 Scatter Diagram or Dot Diagram 239
10.5 Karl Pearson's Coefficient of Correlation 241
10.6 Assumptions 242
(xi)
