Statistical methods / N.G. Das

1. Collection of Data: Classification and Tabulation
1.1 Meaning of 'Statistics' 1
1.2 Variable and Attribute 3
1.3 Primary Data and Secondary Data 4
1.4 Population (or Universe) and Sample 6
1.5 Complete Enumeration (or Census) and Sample Survey 7
1.6 Statistical Enquiry 8
1.7 Useful Terms 11
1.8 Classification 14
1.9 Tabulation 14
1.10 Mechanical Tabulation 20
Exercises 21
Answers 24
2. Charts and Diagrams
2.1 Objects of Diagrammatic Representation 27
2.2 Types of Charts and Diagrams 27
Exercises 41
Answers 43
3. Useful Mathematical Devices
3.1 Rounding of Numbers 44
3.2 Absolute, Relative and Percentage Errors 44
3.3 Significant Figures 45
3.4 Some Short Processes of Calculation 46
3.5 Roots and Reciprocals Expressed as Power 49
3.6 Logarithm 49
3.7 A.P. Series and G.P. Series 54
3.8 Sum and Sum of the Squares of Numbers 55
3.9 Simple Interest Law and Compound Interest Law 55
3.10 Permutation and Combination 56
3.11 Binomial Series and Binomial Coefficients 56
3.12 Inequalities 57
3.13 Concept of'Function' 57
3.14 Polynomial 58
3.15 Sigma (E) Notation 58
3.16 Simple Interpolation 63
4. Frequency Distribution
4.1 Observation, Frequency 66
4.2 Simple Series (or Ungrouped Data) and Frequency Distribution 66
4.3 Useful Terms Associated with Grouped Frequency Distributions 69
4.4 Construction of Frequency Distribution 78
4.5 Cumulative Frequency Distribution 81
4.6 Relative Frequency Distribution 87
4.7 Diagrammatic Representation of Frequency Distributions 87
4.8 Frequency Curve 95
Exercises 97
Answers 99
5. Measures of Central Tendency
5.1 Averages or Measures of Central Tendency lOI
5.2 Arithmetic Mean (A.M.) 106
5.3 Important Properties of A.M. 110
5.4 Simplified Calculation for A.M. 115
5.5 Mean of Composite Group 124
5.6 Geometric Mean (G.M.) 126
5.7 Properties of G.M. 127
5.8 Harmonic Mean (H.M.) 133
5.9 Advantages and Disadvantages of A.M., G.M., H.M. 135
5.10 Relations between A.M., G.M., H.M. 137
5.11 Median 140
5.12 Calculation of Median 140
5.13 Advantages and Disadvantages of Median 141
5.14 Mode 152
5.15 Calculation of Mode 152
5.16 Advantages and Disadvantages of Mode 153
5.17 Relation between Mean, Median, Mode 157
5.18 Partition Values—Quartiles, Deciles, Percentiles 157
5.19 Calculation of Partition Values 158
Exercises 166
Answers 173
6. Measures of Dispersion
6.1 Meaning and Necessity of 'Measures of Dispersion' 175
6.2 Range 179
6.3 Quartile Deviation (or Semi-interquartile Range) 180
6.4 Mean Deviation (or Mean Absolute Deviation) 182
6.5 Standard Deviation (S.D.) 184
6.6 Important Properties of S.D. 185
6.7 Calculation of Standard Deviation (s) 194
6.8 S.D. of Composite Group 203
6.9 Relation between S.D. and Other Measures 208
6.10 Relative Measures of Dispersion 209
6. 1 I I i'l lMl/ ( "lll\ C J "
lixmnrs 21"
.4/jvui/^ 22.^
7. Moments, Skewnes>s and Kiirtosi.s
7 I McimcntN 22-4
7.2 Relation bctuccn CVnir.il aiul Noiv ccniral MomctUs 22S
12 Beia cocllk icnis ami (iainina'CoelfK ienis 22^
7 4 Sianilarili/cd Variable 2Mi
1.2 Motneiiis of iTeqiiency Disiribulions 220
10 Sbeppard's CmreebtMi Ita brmis due u> (iroupmg 222
1 1 Ske\MK*vs 222
7.x Kurtosis 240
Excrcisv\ 24 /
Answers 244
8. Curve Fitting and Method of I.east Squares
X.I Curve bitting 245
X.2 Straight Line and Parabola 240
X.3 l-ree hand Method of Curve r-itting 240
X.4 Method of I-ea.st Squares 240
8.5 Fitting Straight Line 25/
8.6 Simplified Caleulations 25/
8.7 Fitting Parabola 250
8.8 Fitting F.\ponential and CJeometric Curves 202
Exercises 205
Answers 207
Mathematical Note 207
9. Correlation and Regression
9.1 Concepts of'Correlation'and'Regression' 260
9.2 Bivariate Data 260
9.3 Bivariate Frequency Distribution 270
9.4 Scatter Diagram 272
9.5 Correlation 275
9.6 Covariance 275
9.7 Correlation Coefficient (; ) 276
9.8 Properties of Correlation Coefficient 277
9.9 Calculation of r 277
9.10 Interpretation and Use of r 2H5
9.11 Variance of the Sum (Difference) of Two Series 2H6
9.12 Regression 290
9.13 Properties of Linear Regression 29/
9.14 Explained Variation and Unexplained Variation 301
9.15 Regression Curve in Bivariate Frequency Distribution 303
9.16 Rank Correlation 304
9.17 Multiple Correlation and Partial Correlation 308
Exercises 311
Answers 316
10. Interpolation
10.1 Introduction 318
10.2 Finite Differences: D and E Operators 318
10.3 Differences of a Polynomial Function 322
10.4 Newton's Forward Interpolation Formula 326
10.5 Newton's Backward Interpolation Formula 329
10.6 Central Difference Formulae-Stirling's and Bessel's 332
10.7 Lagrange's Interpolation Formula 333
10.8 Inverse Interpolation 335
11. Theory of Probability
11.1 Introduction 343
11.2 Random Experiment, Outcome, Event 343
11.3 Important Terminology 347
11.4 Techniques of Counting 349
11.5 Classical (or 'a Priori') Definition of Probability 351
ri .6 Theorems of Probability 362
11.7 Drawing without Replacement 373
11.8 Repeated Trials—Drawing with Replacement 377
11.9 Bayes' Theorem 379
11.10 Mathematical Expectation 383
11.11 Other Approaches to Probability Theory 387
11.12 Set Theory 388
11.13 Set and Probability 392
11.14 Axioms of Probability 396
11.15 Finite Probability Space and Assignment of Probabilities 398
11.16 Finite Equiprobable Sample Space and Classical Definition 399
11.17 Conditional Probability 401
11.18 Independent Events 402
11.19 Random Variable 406
11.20 Cumulative Distribution Function (C.D.F.) 410
11.21 Joint Distribution of Two Variables (Discrete) 413
Exercises 415
Answers 425
12. Theoretical Distiibutions-Binoinial, Poisson, Normal
12.1 Random Variable and Probability Distribution 426
12.2 Discrete Probability Distribution 426
12.3 Expectation.s—Mean, Variance, Moments (Discrete Distribution) 429
12.4 Uniform Distribution (Discrete) 432
12.5 Binomial Distribution 433
12.6 Poisson Distribution 441
IfJL
7 "" Approvmiation u» Hinointal Disirilnilmn 44^
I2.S n\pcii!c»>nicliic 1 )isirihiiliiMJ 447
12,'' Miilimoini.il Disiribuiioii
2 10 Dismbulion nl Iwi* \an;ibk's 4>l
12 1 1 ( ontiiuioiis Probabilil> Distnbulinn 4ft5
2.12 I■mtoriii Dislnbiuiiin (('oniimioiis) 467
2 ! ^ Noimal Distnbiilion 4fhS
2. 14 Norin.il Appro\imatii)n in l^inntniai (PoissoiO 475
2. 15 ("ntiirni I.irnii Ihcoivin 47"
I-.\tni\t"t 477
Answers 4Sft
13. Sampling Theory
I .VI Meaning aiul Objecls ol 'Sampling" 4cV<S'
I .V2 Sampling Faror and BIAs 4^)0
I .V.^ lypes ol" Sampling 492
I .V4 Mcihod ol' Drawing Random Sample 495
I.V5 Sampling Distribution 499
I.V6 Two Impoitant Sampling Distribution (l.argc Sample) 501
I.V7 Slaiuiard f'irror (S.IV) 501
I .V7A ITobable lirror (l^.li.} 5/6
I3.S Distributions Used in Sampling Theory 5/7
A.ve/c/.ve.v 525
Answers 526
14. Estimation and Test of Significance
14.1 Introduction 52H
14.2 Theory ol" Kstimation 52H
14.3 Point Estimation—Criteria for Good Estimators 52H
14.4 Methods of Point Estimation 554
14.5 Interval Estimation 557
14.6 Theory of Test of Significance 546
14.7 Large Sample Te.sts 554
14.8 Small Sample Tests 572
14.9 Tests for Correlation Coefficient 5H7
Exercises 601
Answers 611
15. Analysis of Variance
15. 1 Introduction 6/5
15.2 Different Sources of Variation 615
15.3 Technique in One-way Classified Data 616
15.4 Steps in Computation (One-way Cla.ssified Data) 618
15.5 Locating Unequal Pairs of Means 619
15.6 Technique in Two-way Classified Data 623
15.7 Steps in Computation (Two-way Classified Data) 625
Exercises 627
Answers 630
16. Time Series
16.1 Meaning and Necessity of'Time Series Analysis' 632
16.2 Components of Time Series 632
16.3 Adjustments to Time Series Data 634
16.4 Secular Trend 635
16.5 Measurement of Trend 635
16.6 Monthly Trend from Annual Data 653
16.7 Seasonal Variation 658
16.8 Measurement of Seasonal Variation 661
16.9 Cyclical Fluctuation 672
16.10 Business Forecasting 673
16.11 Exponential Smoothing 674
Exercises 676
Answers 683
17. Index Numbers
17.1 Meaning of'Index Number' 686
17.2 Problems in Construction of Index Numbers 688
17.3 Methods of Construction of Index Numbers 689
17.4 Quantity Index Number 701
17.5 Tests of Index Numbers 706
17.6 Chain Base Method 713
n.l Cost of Living Index Numbers 715
17.8 Bias in Laspeyres' and Paasche's Formulae for C.L.I. 724
17.9 Base Shifting, Splicing and Deflation 726
17.10 Errors in Index Numbers 730
Exercises 730
Answers 741
18. Vital Statistics
18.1 Introduction 743
18.2 Crude Death Rate 744
18.3 Specific Death Rate 744
18.4 Standardised Death Rate 745
18.5 Life Table 748
18.6 Crude Birth Rate 752
18.7 General Fertility Rate 752
18.8 Age-specific Fertility Rate 753
18.9 Total Fertility Rate 753
18.10 Vital Index 753
18.11 Gross Reproduction Rate 754
18.12 Net Reproduction Rate 754
Exercises 758
Answers 763
19. Statistical Quality Control
19.1 Introduction 765
19.2 Chance Causes and Assignable Causes 766
19.3 Control Chart—How It Works 766
19.4 Control Charts for Variables and Attributes 768
19.5 Formulae for Central Line and Control Limits 769
19.6 Sampling Inspection 777
19.7 Single and Double Sampling Inspection Plans 778
19.8 Important Terms used in Sampling Inspection 779
Exercises 780
Answers 783