Appendix B - Statistical Tables

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APPENDIX B Statistical Tables

TABLE B.1 THE UNIT NORMAL TABLE*

*Column A lists z-score values. A vertical line drawn through a normal distribution at a z-score location divides the

distribution into two sections.

Column B identifies the proportion in the larger section, called the body.

Column C identifies the proportion in the smaller section, called the tail.

Column D identifies the proportion between the mean and the z-score.

Note: Because the normal distribution is symmetrical, the proportions for negative z-scores are the same as those for

positive z-scores.

0 +z -z 0 0 z

(A) (B) (C) (D) (A) (B) (C ) (D)

Proportion Proportion Proportion Proportion Proportion Proportion

z in Body in Tail Between Mean and z z in Body in Tail Between Mean and z

0.00 .5000 .5000 .0000 0.25 .5987 .4013 .0987

0.01 .5040 .4960 .0040 0.26 .6026 .3974 .1026

0.02 .5080 .4920 .0080 0.27 .6064 .3936 .1064

0.03 .5120 .4880 .0120 0.28 .6103 .3897 .1103

0.04 .5160 .4840 .0160 0.29 .6141 .3859 .1141

0.05 .5199 .4801 .0199 0.30 .6179 .3821 .1179

0.06 .5239 .4761 .0239 0.31 .6217 .3783 .1217

0.07 .5279 .4721 .0279 0.32 .6255 .3745 .1255

0.08 .5319 .4681 .0319 0.33 .6293 .3707 .1293

0.09 .5359 .4641 .0359 0.34 .6331 .3669 .1331

0.10 .5398 .4602 .0398 0.35 .6368 .3632 .1368

0.11 .5438 .4562 .0438 0.36 .6406 .3594 .1406

0.12 .5478 .4522 .0478 0.37 .6443 .3557 .1443

0.13 .5517 .4483 .0517 0.38 .6480 .3520 .1480

0.14 .5557 .4443 .0557 0.39 .6517 .3483 .1517

0.15 .5596 .4404 .0596 0.40 .6554 .3446 .15540.16 .5636 .4364 .0636 0.41 .6591 .3409 .1591

0.17 .5675 .4325 .0675 0.42 .6628 .3372 .1628

0.18 .5714 .4286 .0714 0.43 .6664 .3336 .1664

0.19 .5753 .4247 .0753 0.44 .6700 .3300 .1700

0.20 .5793 .4207 .0793 0.45 .6736 .3264 .1736

0.21 .5832 .4168 .0832 0.46 .6772 .3228 .1772

0.22 .5871 .4129 .0871 0.47 .6808 .3192 .1808

0.23 .5910 .4090 .0910 0.48 .6844 .3156 .1844

0.24 .5948 .4052 .0948 0.49 .6879 .3121 .1879



726 APPENDIX B STATISTICALTABLES

(A) (B)

Proportion

z in Body

(C)

Proportion

inTail

(D)

Proportion

Between Mean and z

(A)

z

(B)

Proportion

inBody

(C)

Proportion

inTail

(D)

Proportion

Between Mean and z

0.50 .6915

0.51 .6950

0.52 .6985

0.53 .7019

0.54 .7054

0.55 .7088

0.56 .7123

0.57 .7157

0.58 .7190

0.59 .7224

0.60 .7257

0.61 .7291

0.62 .7324

0.63 .7357

0.64 .7389

0.65 .7422

0.66 .7454

0.67 .74860.68 .7517

0.69 .7549

0.70 .7580

0.71 .7611

0.72 .7642

0.73 .7673

0.74 .7704

0.75 .7734

0.76 .7764

0.77 .7794

0.78 .7823

0.79 .7852

0.80 .7881

0.81 .7910

0.82 .7939

0.83 .7967

0.84 .7995

0.85 .8023

0.86 .8051

0.87 .8078

0.88 .8106

0.89 .8133

0.90 .8159

0.91 .8186

0.92 .8212

0.93 .8238

0.94 .82640.95 .8289

0.96 .8315

0.97 .8340

0.98 .8365

0.99 .8389

.3085

.3050

.3015

.2981

.2946

.2912

.2877

.2843

.2810

.2776

.2743

.2709

.2676

.2643

.2611

.2578

.2546

.2514

.2483

.2451

.2420

.2389

.2358

.2327

.2296

.2266

.2236

.2206

.2177

.2148

.2119

.2090

.2061

.2033

.2005

.1977

.1949

.1922

.1894

.1867

.1841

.1814

.1788

.1762

.1736

.1711

.1685

.1660

.1635

.1611

.1915

.1950

.1985

.2019

.2054

.2088

.2123

.2157

.2190

.2224

.2257

.2291

.2324

.2357

.2389

.2422

.2454

.2486

.2517

.2549

.2580

.2611

.2642

.2673

.2704

.2734

.2764

.2794

.2823

.2852

.2881

.2910

.2939

.2967

.2995

.3023

.3051

.3078

.3106

.3133

.3159

.3186

.3212

.3238

.3264

.3289

.3315

.3340

.3365

.3389

1.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1 . 1 0

1 . 1 1

1 . 1 2

1 . 1 3

1 . 1 4

1 . 1 5

1 . 1 6

1 . 1 7

1 . 1 8

1 . 1 9

1.20

1.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1 . 3 0

1 . 3 1

1 . 3 2

1 . 3 3

1 . 3 4

1 . 3 5

1 . 3 6

1 . 3 7

1 . 3 8

1 . 3 9

1.40

1.41

1.42

1.43

1.441.45

1.46

1.47

1.48

1.49

.8413

.8438

.8461

.8485

.8508

.8531

.8554

.8577

.8599

.8621

.8643

.8665

.8686

.8708

.8729

.8749

.8770

.8790

.8810

.8830

.8849

.8869

.8888

.8907

.8925

.8944

.8962

.8980

.8997

.9015

.9032

.9049

.9066

.9082

.9099

.9115

.9131

.9147

.9162

.9177

.9192

.9207

.9222

.9236

.9251

.9265

.9279

.9292

.9306

.9319

.1587

.1562

.1539

.1515

.1492

.1469

.1446

.1423

.1401

.1379

.1357

.1335

.1314

.1292

.1271

.1251

.1230

.1210

.1190

.1170

.1151

.1131

.1112

.1093

.1075

.1056

.1038

.1020

.1003

.0985

.0968

.0951

.0934

.0918

.0901

.0885

.0869

.0853

.0838

.0823

.0808

.0793

.0778

.0764

.0749

.0735

.0721

.0708

.0694

.0681

.3413

.3438

.3461

.3485

.3508

.3531

.3554

.3577

.3599

.3621

.3643

.3665

.3686

.3708

.3729

.3749

.3770

.3790

.3810

.3830

.3849

.3869

.3888

.3907

.3925

.3944

.3962

.3980

.3997

.4015

.4032

.4049

.4066

.4082

.4099

.4115

.4131

.4147

.4162

.4177

.4192

.4207

.4222

.4236

.4251

.4265

.4279

.4292

.4306

.4319



APPEND IX B STAT IST ICAL TABLES

(A) (B)

Proportion

z in Body

1.50 .9332

1.51 .9345

1.52 .9357

1.53 .9370

1.54 .9382

1.55 .9394

1.56 .9406

1.57 .9418

1.58 .9429

1.59 .9441

1.60 .9452

1.61 .9463

1.62 .9474

1.63 .9484

1.64 .9495

1.65 .9505

1.66 .9515

1.67 .95251.68 .9535

1.69 .9545

1.70 .9554

1.71 .9564

1.72 .9573

1.73 .9582

1.74 .9591

1.75 .9599

1.76 .9608

1.77 .9616

1.78 .9625

1.79 .9633

1.80 .9641

1.81 .9649

1.82 .9656

1.83 .9664

1.84 .9671

1.85 .9678

1.86 .9686

1.87 .9693

1.88 .9699

1.89 .9706

1.90 .9713

1.91 .9719

1.92 .9726

1.93 .9732

1.94 .9738

1.95 .9744

1.96 .9750

1.97 .9756

1.98 .9761

1.99 .9767

(C)

Proportion

inTail

.0668

.0655

.0643

.0630

.0618

.0606

.0594

.0582

.0571

.0559

.0548

.0537

.0526

.0516

.0505

.0495

.0485

.0475

.0465

.0455

.0446

.0436

.0427

.0418

.0409

.0401

.0392

.0384

.0375

.0367

.0359

.0351

.0344

.0336

.0329

.0322

.0314

.0307

.0301

.0294

.0287

.0281

.0274

.0268

.0262

.0256

.0250

.0244

.0239

.0233

(D)

Proportion

Between Mean and z

.4332

.4345

.4357

.4370

.4382

.4394

.4406

.4418

.4429

.4441

.4452

.4463

.4474

.4484

.4495

.4505

.4515

.4525

.4535

.4545

.4554

.4564

.4573

.4582

.4591

.4599

.4608

.4616

.4625

.4633

.4641

.4649

.4656

.4664

.4671

.4678

.4686

.4693

.4699

.4706

.4713

.4719

.4726

.4732

.4738

.4744

.4750

.4756

.4761

.4767

(A) (B)

Proportion

z inBody

2.00 .9772

2.01 .9778

2.02 .9783

2.03 .9788

2.04 .9793

2.05 .9798

2.06 .9803

2.07 .9808

2.08 .9812

2.09 .9817

2.10 .9821

2.11 .9826

2.12 .9830

2.13 .9834

2.14 .9838

2.15 .9842

2.16 .9846

2.17 .98502.18 .9854

2.19 .9857

2.20 .9861

2.21 .9864

2.22 .9868

2.23 .9871

2.24 .9875

2.25 .9878

2.26 .9881

2.27 .9884

2.28 .9887

2.29 .9890

2.30 .9893

2.31 .9896

2.32 .9898

2.33 .9901

2.34 .9904

2.35 .9906

2.36 .9909

2.37 .9911

2.38 .9913

2.39 .9916

2.40 .9918

2.41 .9920

2.42 .9922

2.43 .9925

2.44 .9927

2.45 .9929

2.46 .9931

2.47 .9932

2.48 .9934

2.49 .9936

(C)

Proportion

inTail

(D)

Proportion

Between Mean and z

.0228

.0222

.0217

.0212

.0207

.0202

.0197

.0192

.0188

.0183

.0179

.0174

.0170

.0166

.0162

.0158

.0154

.0150

.0146

.0143

.0139

.0136

.0132

.0129

.0125

.0122

.0119

.0116

.0113

.0110

.0107

.0104

.0102

.0099

.0096

.0094

.0091

.0089

.0087

.0084

.0082

.0080

.0078

.0075

.0073

.0071

.0069

.0068

.0066

.0064

.4772

.4778

.4783

.4788

.4793

.4798

.4803

.4808

.4812

.4817

.4821

.4826

.4830

.4834

.4838

.4842

.4846

.4850

.4854

.4857

.4861

.4864

.4868

.4871

.4875

.4878

.4881

.4884

.4887

.4890

.4893

.4896

.4898

.4901

.4904

.4906

.4909

.4911

.4913

.4916

.4918

.4920

.4922

.4925

.4927

.4929

.4931

.4932

.4934

.4936




(A) (B)

Proportion

z in Body

(C)

Proportion

inTail

(D)

Proportion

Between Mean and z

(A)

z

(B)

Proportion

inBody

(C)

Proportion

inTail

(D)

Proportion

Between Mean and z

2.50 .9938

2.51 .9940

2.52 .9941

2.53 .9943

2.54 .9945

2.55 .9946

2.56 .9948

2.57 .9949

2.58 .9951

2.59 .9952

2.60 .9953

2.61 .9955

2.62 .9956

2.63 .9957

2.64 .9959

2.65 .9960

2.66 .9961

2.67 .99622.68 .9963

2.69 .9964

2.70 .9965

2.71 .9966

2.72 .9967

2.73 .9968

2.74 .9969

2.75 .9970

2.76 .9971

2.77 .9972

2.78 .9973

2.79 .9974

2.80 .9974

2.81 .9975

2.82 .9976

2.83 .9977

2.84 .9977

2.85 .9978

2.86 .9979

2.87 .9979

2.88 .9980

2.89

2.90

2.91

2.92

2.93

2.94

.9981

.9981

.9982

.9982

.9983

.9984

.0062

.0060

.0059

.0057

.0055

.0054

.0052

.0051

.0049

.0048

.0047

.0045

.0044

.0043

.0041

.0040

.0039

.0038

.0037

.0036

.0035

.0034

.0033

.0032

.0031

.0030

.0029

.0028

.0027

.0026

.0026

.0025

.0024

.0023

.0023

.0022

.0021

.0021

.0020

.0019

.0019

.0018

.0018

.0017

.0016

.4938

.4940

.4941

.4943

.4945

.4946

.4948

.4949

.4951

.4952

.4953

.4955

.4956

.4957

.4959

.4960

.4961

.4962

.4963

.4964

.4965

.4966

.4967

.4968

.4969

.4970

.4971

.4972

.4973

.4974

.4974

.4975

.4976

.4977

.4977

.4978

.4979

.4979

.4980

.4981

.4981

.4982

.4982

.4983

.4984

2.95

2.96

2.97

2.98

2.99

3.00

3.01

3.02

3.03

3.04

3.05

3.06

3.07

3.08

3.09

3.10

3.11

3.123.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.30

3.40

3.50

3.60

3.70

3.80

3.90

4.00

.9984

.9985

.9985

.9986

.9986

.9987

.9987

.9987

.9988

.9988

.9989

.9989

.9989

.9990

.9990

.9990

.9991

.9991

.9991

.9992

.9992

.9992

.9992

.9993

.9993

.9993

.9993

.9994

.9994

.9994

.9995

.9997

.9998

.9998

.9999

.99993

.99995

.99997

.0016

.0015

.0015

.0014

.0014

.0013

.0013

.0013

.0012

.0012

.0011

.0011

.0011

.0010

.0010

.0010

.0009

.0009

.0009

.0008

.0008

.0008

.0008

.0007

.0007

.0007

.0007

.0006

.0006

.0006

.0005

.0003

.0002

.0002

.0001

.00007

.00005

.00003

.4984

.4985

.4985

.4986

.4986

.4987

.4987

.4987

.4988

.4988

.4989

.4989

.4989

.4990

.4990

.4990

.4991

.4991

.4991

.4992

.4992

.4992

.4992

.4993

.4993

.4993

.4993

.4994

.4994

.4994

.4995

.4997

.4998

.4998

.4999

.49993

.49995

.49997




TABLE B .2 THE t DISTRIBUTION

Table entries are values of t corresponding to proportions in one tailor in two tails combined.

One tail

(either right or left)

Two tails

combined

Proportion in One Tail

0.25 0.10 0.05 0.025 0.01 0.005

Proportion in Two Tails Combined

df 0.50 0.20 0.10 0.05 0.02 0.01

1.000 3.078 6.314 12.706 31.821 63.657

2 0.816 1.886 2.920 4.303 6.965 9.925

3 0.765 1.638 2.353 3.182 4.541 5.841

4 0.741 1.533 2.132 2.776 3.747 4.604

5 0.727 1.476 2.015 2.571 3.365 4.032

6 0.718 1.440 1.943 2.447 3.143 3.707

7 0.711 1.415 1.895 2.365 2.998 3.499

8 0.706 1.397 1.860 2.306 2.896 3.355

9 0.703 1.383 1.833 2.262 2.821 3.250

10 0.700 1.372 1.812 2.228 2.764 3.169

11 0.697 1.363 1.796 2.201 2.718 3.106

12 0.695 1.356 1.782 2.179 2.681 3.055

13 0.694 1.350 1.771 2.160 2.650 3.012

14 0.692 1.345 1.761 2.145 2.624 2.97715 0.691 1.341 1.753 2.131 2.602 2.947

16 0.690 1.337 1.746 2.120 2.583 2.921

17 0.689 1.333 1.740 2.110 2.567 2.898

18 0.688 1.330 1.734 2.101 2.552 2.878

19 0.688 1.328 1.729 2.093 2.539 2.861

20 0.687 1.325 1.725 2.086 2.528 2.845

21 0.686 1.323 1.721 2.080 2.518 2.831

22 0.686 1.321 1.717 2.074 2.508 2.819

23 0.685 1.319 1.714 2.069 2.500 2.807

24 0.685 1.318 1.711 2.064 2.492 2.797

25 0.684 1.316 1.708 2.060 2.485 2.787

26 0.684 1.315 1.706 2.056 2.479 2.779

27 0.684 1.314 1.703 2.052 2.473 2.771

28 0.683 1.313 1.701 2.048 2.467 2.76329 0.683 1.311 1.699 2.045 2.462 2.756

30 0.683 1.310 1.697 2.042 2.457 2.750

40 0.681 1.303 1.684 2.021 2.423 2.704

60 0.679 1.296 1.671 2.000 2.390 2.660

120 0.677 1.289 1.658 1.980 2.358 2.617

00 0.674 1.282 1.645 1.960 2.326 2.576

Table III of R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research, 6th ed. London: Longman Group Lt

1974 (previously published by Oliver and Boyd Ltd., Edinburgh). Copyright ©1963 R. A. Fisher and F. Yates. Adapted and reprinted with

permission of Pearson Education Limited.



730 A PPE ND IX B ST ATIST IC AL TA BLE S

TABLE B.3 CRITICAL VALUES FOR THE F-MAX STATISTIC*

*The critical values for ex= .05 are in lightface type, and for ex= .01, they are in boldface type.

k =Number of Samples

n - 1 2 3 4 5 6 7 8 9 10 11 12

4 9.60 15.5 20.6 25.2 29.5 33.6 37.5 41.4 44.6 48.0 51.4

23.2 37. 49. 59. 69. 79. 89. 97. 106. 113. 120.

5 7.15 10.8 13.7 16.3 18.7 20.8 22.9 24.7 26.5 28.2 29.9

14.9 22. 28. 33. 38. 42. 46. 50. 54. 57. 60.

6 5.82 8.38 10.4 12.1 13.7 15.0 16.3 17.5 18.6 19.7 20.7

11.1 15.5 19.1 22. 25. 27. 30. 32. 34. 36. 37.

7 4.99 6.94 8.44 9.70 10.8 11.8 12.7 13.5 14.3 15.1 15.8

8.89 12.1 14.5 16.5 18.4 20. 22. 23. 24. 26. 27.

8 4.43 6.00 7.18 8.12 9.03 9.78 10.5 11.1 11.7 12.2 12.7

7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 19.8 21.

9 4.03 5.34 6.31 7.11 7.80 8.41 8.95 9.45 9.91 10.3 10.7

6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 16.0 16.6

10 3.72 4.85 5.67 6.34 6.92 7.42 7.87 8.28 8.66 9.01 9.34

5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 13.4 13.9

12 3.28 4.16 4.79 5.30 5.72 6.09 6.42 6.72 7.00 7.25 7.48

4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 10.2 10.6

15 2.86 3.54 4.01 4.37 4.68 4.95 5.19 5.40 5.59 5.77 5.93

4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 7.8 8.0

20 2.46 2.95 3.29 3.54 3.76 3.94 4.10 4.24 4.37 4.49 4.59

3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 5.8 5.9

30 2.07 2.40 2.61 2.78 2.91 3.02 3.12 3.21 3.29 3.36 3.39

2.63 3.0 3.3 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2

60 1.67 1.85 1.96 2.04 2.11 2.17 2.22 2.26 2.30 2.33 2.36

1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 2.7 2.7

Table 31 ofE. Pearson and H. O. Hartley, Biometrika Tablesfor Statisticians, 2nd ed. New York: Cambridge University Press, 1958. Adapted and

reprinted with permission of the Biometrika trustees.



TABLE B .4 THE F DISTRIBUTION*


*Table entries in lightface type are critical values for the .05 level of significance.

Boldface type values are for the .01 level of significance.

Critical

F

Degrees of

Freedom:

Denominator 2 3 4 5 6

Degrees of Freedom: Numerator

7 8 9 10 11 12 14 16 2

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

161 200 2164052 4999 5403

2255625

230 2345764 5859

2375928

18.51 19.00 19.16 19.25 19.30 19.33 19.36

98.49 99.00 99.17 99.25 99.30 99.33 99.34

10.13 9.55 9.28 9.12 9.01 8.94 8.88

34.12 30,92 29.46 28.71 28.24 27.91 27.67

2395981

19.37 19.38

99.36 99.38

8.84 8.81

27.49 27.34

6.04

14.80

4.82

10.27

4.15

8.10

3.73

6.84

3.44

6.03

3.23

5.47

3.07

5.06

2.95

4.74

2.85

4.50

2.77

4.30

2.70

4.14

2.64

4.00

2.59

3.89

2.55

3.79

2.51

3.71

6.00

14.66

4.78

10.15

4.10

7.98

3.68

6.71

3.39

5.91

3.18

5.35

3.02

4.95

2.90

4.63

2.80

4.39

2.72

4.19

2.65

4.03

2.59

3.89

2.54

3.78

2.50

3.68

2.46

3.60

2416022

2426056

7.71 6.94 6.59 6.39 6.26 6.16

21.20 18.00 16.69 15.98 15.52 15.21

6.61 5.79 5.41 5.19 5.05 4.95

16.26 13.27 12.06 11.39 10.97 10.67

5.99 5.14 4.76 4.53 4.39 4.28

13.74 10,92 9.78 9.15 8.75 8.47

5.59 4.74 4.35 4.12 3.97 3.87

12.25 9.55 8.45 7.85 7.46 7.19

5.32 4.46 4.07 3.84 3.69 3.58

11.26 8.65 7.59 7.01 6.63 6.37

5.12 4.26 3.86 3.63 3.48 3.37

10.56 8.02 6.99 6.42 6.06 5.80

4.96 4.10 3.71 3.48 3.33 3.22

10.04 7.56 6.55 5.99 5.64 5.39

4.84 3.98 3.59 3.36 3.20 3.09

9.65 7.20 6.22 5.67 5.32 5.07

4.75 3.88 3.49 3.26 3.11 3.00

9.33 6.93 5.95 5.41 5.06 4.82

4.67 3.80 3.41 3.18 3.02 2.92

9.07 6.70 5.74 5.20 4.86 4.62

4.60 3.74 3.34 3.11 2.96 2.85

8.86 6.51 5.56 5.03 4.69 4.46

4.54 3.68 3.29 3.06 2.90 2.79

8.68 6.36 5.42 4.89 4.56 4.32

4.49 3.63 3.24 3.01 2.85 2.74

8.53 6.23 5.29 4.77 4.44 4.20

4.45 3.59 3.20 2.96 2.81 2.70

8.40 6.11 5.18 4.67 4.34 4.10

4.41 3.55 3.16 2.93 2.77 2.66

8.28 6.01 5.09 4.58 4.25 4.01

6.09

14.98

4.88

10.45

4.21

8.26

3.79

7.00

3.50

6.19

3.29

5.62

3.14

5.21

3.01

4.88

2.92

4.65

2.84

4.44

2.77

4.28

2.70

4.14

2.66

4.03

2.62

3.93

2.58

3.85

2436082

244 245 246 26106 6142 6169 62

19.39 19.40 19.41 19.42 19.43 19

99.40 99.41 99.42 99.43 99.44 99

8.78 8.76 8.74 8.71 8.69 8

27.23 27.13 27.05 26.92 26.83 26

5.96 5.93 5.91 5.87 5.84 5

14.54 14.45 14.37 14.24 14.15 14

4.74 4.70 4.68 4.64 4.60 4

10.05 9.96 9.89 9.77 9.68 9

4.06 4.03 4.00 3.96 3.92 3

7.87 7.79 7.72 7.60 7.52 7

3.63 3.60 3.57 3.52 3.49 3

6.62 6.54 6.47 6.35 6.27 6

3.34 3.31 3.28 3.23 3.20 3

5.82 5.74 5.67 5.56 5.48 5

3.13 3.10 3.07 3.02 2.98 2

5.26 5.18 5.11 5.00 4.92 4

2.97 2.94 2.91 2.86 2.82 2

4.85 4.78 4.71 4.60 4.52 4

2.86 2.82 2.79 2.74 2.70 2

4.54 4.46 4.40 4.29 4.21 4

2.76 2.72 2.69 2.64 2.60 2

4.30 4.22 4.16 4.05 3.98 3

2.67 2.63 2.60 2.55 2.51 2

4.10 4.02 3.96 3.85 3.78 3

2.60 2.56 2.53 2.48 2.44 2

3.94 3.86 3.80 3.70 3.62 3

2.55 2.51 2.48 2.43 2.39 2

3.80 3.73 3.67 3.56 3.48 3

2.49 2.45 2.42 2.37 2.33 2

3.69 3.61 3.55 3.45 3.37 3

2.45 2.41 2.38 2.33 2.29 2

3.59 3.52 3.45 3.35 3.27 3

2.41 2.37 2.34 2.29 2.25 2

3.51 3.44 3.37 3.27 3.19 3




TABLE B .4 (continued)

Degrees of

Freedom:

Denominator 2 3 4 5 6


98 10 11 12 14 16 20

19

20

21

22

23

24

25

26

27

28

29

30

32

34

36

38

40

42

4.38

8.18

4.35

8.10

4.32

8.02

4.30

7.94

4.28

7.88

4.26

7.82

4.24

7.77

4.227.72

4.21

7.68

4.20

7.64

4.18

7.60

4.17

7.56

4.15

7.50

4.13

7.44

4.11

7.39

4.10

7.35

4.08

7.31

4.07

7.27

3.52 3.13

5.93 5.01

3.49 3.10

5.85 4.94

3.47 3.07

5.78 4.87

3.44 3.05

5.72 4.82

3.42 3.03

5.66 4.76

3.40 3.01

5.61 4.72

3.38 2.99

5.57 4.68

3.37 2.985.53 4.64

3.35 2.96

5.49 4.60

3.34 2.95

5.45 4.57

3.33 2.93

5.42 4.54

3.32 2.92

5.39 4.51

3.30 2.90

5.34 4.46

3.28 2.88

5.29 4.42

3.26 2.86

5.25 4.38

3.25 2.85

5.21 4.34

3.23 2.84

5.18 4.31

3.22 2.83

5.15 4.29

2.90

4.50

2.87

4.43

2.84

4.37

2.82

4.31

2.80

4.26

2.78

4.22

2.76

4.18

2.744.14

2.73

4.11

2.71

4.07

2.70

4.04

2.69

4.02

2.67

3.97

2.65

3.93

2.63

3.89

2.62

3.86

2.61

3.83

2.59

3.80

2.74

4.17

2.71

4.10

2.68

4.04

2.66

3.99

2.64

3.94

2.62

3.90

2.60

3.86

2.593.82

2.57

3.79

2.56

3.76

2.54

3.73

2.53

3.70

2.51

3.66

2.49

3.61

2.48

3.58

2.46

3.54

2.45

3.51

2.44

3.49

2.63

3.94

2.60

3.87

2.57

3.81

2.55

3.76

2.53

3.71

2.51

3.67

2.49

3.63

2.473.59

2.46

3.56

2.44

3.53

2.43

3.50

2.42

3.47

2.40

3.42

2.38

3.38

2.36

3.35

2.35

3.32

2.34

3.29

2.32

3.26

2.55

3.77

2.52

3.71

2.49

3.65

2.47

3.59

2.45

3.54

2.43

3.50

2.41

3.46

2.393.42

2.37

3.39

2.36

3.36

2.35

3.33

2.34

3.30

2.32

3.25

2.30

3.21

2.28

3.18

2.26

3.15

2.25

3.12

2.24

3.10

2.48

3.63

2.45

3.56

2.42

3.51

2.40

3.45

2.38

3.41

2.36

3.36

2.34

3.32

2.323.29

2.30

3.26

2.29

3.23

2.28

3.20

2.27

3.17

2.25

3.12

2.23

3.08

2.21

3.04

2.19

3.02

2.18

2.99

2.17

2.96

2.43

3.52

2.40

3.45

2.37

3.40

2.35

3.35

2.32

3.30

2.30

3.25

2.28

3.21

2.273.17

2.25

3.14

2.24

3.11

2.22

3.08

2.21

3.06

2.19

3.01

2.17

2.97

2.15

2.94

2.14

2.91

2.12

2.88

2.11

2.86

2.38

3.43

2.35

3.37

2.32

3.31

2.30

3.26

2.28

3.21

2.26

3.17

2.24

3.13

2.223.09

2.20

3.06

2.19

3.03

2.18

3.00

2.16

2.98

2.14

2.94

2.12

2.89

2.10

2.86

2.09

2.82

2.07

2.80

2.06

2.77

2.34

3.36

2.31

3.30

2.28

3.24

2.26

3.18

2.24

3.14

2.22

3.09

2.20

3.05

2.183.02

2.16

2.98

2.15

2.95

2.14

2.92

2.12

2.90

2.10

2.86

2.08

2.82

2.06

2.78

2.05

2.75

2.04

2.73

2.02

2.70

2.31

3.30

2.28

3.23

2.25

3.17

2.23

3.12

2.20

3.07

2.18

3.03

2.16

2.99

2.152.96

2.13

2.93

2.12

2.90

2.10

2.87

2.09

2.84

2.07

2.80

2.05

2.76

2.03

2.72

2.02

2.69

2.00

2.66

1.99

2.64

2.26 2.21 2.15

3.19 3.12 3.00

2.23 2.18 2.12

3.13 3.05 2.94

2.20 2.15 2.09

3.07 2.99 2.88

2.18 2.13 2.07

3.02 2.94 2.83

2.14 2.10 2.04

2.97 2.89 2.78

2.13 2.09 2.02

2.93 2.85 2.74

2.11 2.06 2.00

2.89 2.81 2.70

2.10 2.05 1.992.86 2.77 2.66

2.08 2.03 1.97

2.83 2.74 2.63

2.06 2.02 1.96

2.80 2.71 2.60

2.05 2.00 1.94

2.77 2.68 2.57

2.04 1.99 1.93

2.74 2.66 2.55

2.02 1.97 1.91

2.70 2.62 2.51

2.00 1.95 1.89

2.66 2.58 2.47

1.98 1.93 1.87

2.62 2.54 2.43

1.96 1.92 1.85

2.59 2.51 2.40

1.95 1.90 1.84

2.56 2.49 2.37

1.94 1.89 1.82

2.54 2.46 2.35



TABLE B .4 (continued)

12


6


9

2.23

3.07

2.22

3.05

2.21

3.04

2.20

3.02

2.18

2.98

2.17

2.95

2.15

2.93

2.142.91

2.12

2.87

2.10

2.82

2.08

2.79

2.07

2.76

2.05

2.73

2.03

2.69

2.02

2.66

2.01

2.64

8

2.16

2.94

2.14

2.92

2.14

2.90

2.13

2.88

2.11

2.85

2.10

2.82

2.08

2.79

2.072.77

2.05

2.74

2.03

2.69

2.01

2.65

2.00

2.62

1.98

2.60

1.96

2.55

1.95

2.53

1.94

2.51

2.10

2.84

2.09

2.82

2.08

2.80

2.07

2.78

2.05

2.75

2.04

2.72

2.02

2.70

2.012.67

1.99

2.64

1.97

2.59

1.95

2.56

1.94

2.53

1.92

2.50

1.90

2.46

1.89

2.43

1.88

2.41

10

2.05

2.75

2.04

2.73

2.03

2.71

2.02

2.70

2.00

2.66

1.99

2.63

1.98

2.61

1.972.59

1.95

2.55

1.92

2.51

1.90

2.47

1.89

2.44

1.87

2.41

1.85

2.37

1.84

2.34

1.83

2.32

11

2.01

2.68

2.00

2.66

1.99

2.64

1.98

2.62

1.97

2.59

1.95

2.56

1.94

2.54

1.932.51

1.91

2.48

1.88

2.43

1.86

2.40

1.85

2.37

1.83

2.34

1.81

2.29

1.80

2.26

1.79

2.24

44

46

48

50

55

60

65

70

80

100

125

150

200

400

2

4.06 3.21

7.24 5.12 4.26

4.05 3.20 2.81

7.21 5.10 4.24

4.04 3.19 2.80

7.19 5.08 4.22

4.03 3.18 2.79

7.17 5.06 4.20

4.02 3.17 2.78

7.12 5.01 4.16

4.00 3.15 2.76

7.08 4.98 4.13

3.99 3.14 2.75

7.04 4.95 4.10

3.98 3.13 2.747.01 4.92 4.08

3.96 3.11 2.72

6.96 4.88 4.04

3.94 3.09 2.70

6.90 4.82 3.98

3.92 3.07 2.68

6.84 4.78 3.94

3.91 3.06 2.67

6.81 4.75 3.91

3.89 3.04 2.65

6.76 4.71 3.88

3.86 3.02 2.62

6.70 4.66 3.83

3.85 3.00 2.61

6.66 4.62 3.80

3.84 2.99 2.60

6.64 4.60 3.78

Degrees of

Freedom:

Denominator 3 4 5 14 16 2

1000

2.82 2.58

3.78

2.57

3.76

2.56

3.74

2.56

3.72

2.54

3.68

2.52

3.65

2.51

3.62

2.503.60

2.48

3.56

2.46

3.51

2.44

3.47

2.43

3.44

2.41

3.41

2.39

3.36

2.38

3.34

2.37

3.32

2.43

3.46

2.42

3.44

2.41

3.42

2.40

3.41

2.38

3.37

2.37

3.34

2.36

3.31

2.353.29

2.33

3.25

2.30

3.20

2.29

3.17

2.27

3.14

2.26

3.11

2.23

3.06

2.22

3.04

2.21

3.02

2.31

3.24

2.30

3.22

2.30

3.20

2.29

3.18

2.27

3.15

2.25

3.12

2.24

3.09

2.233.07

2.21

3.04

2.19

2.99

2.17

2.95

2.16

2.92

2.14

2.90

2.12

2.85

2.10

2.82

2.09

2.80

1.98

2.62

1.97

2.60

1.96

2.58

1.95

2.56

1.93

2.53

1.92

2.50

1.90

2.47

1.892.45

1.88

2.41

1.85

2.36

1.83

2.33

1.82

2.30

1.80

2.28

1.78

2.23

1.76

2.20

1.75

2.18

1.92 1.88 1

2.52 2.44 2

1.91 1.87 1

2.50 2.42 2

1.90 1.86 1

2.48 2.40 2

1.90 1.85 1

2.46 2.39 2

1.88 1.83 1

2.43 2.35 2

1.86 1.81 1

2.40 2.32 2

1.85 1.80 1

2.37 2.30 2

1.84 1.79 12.35 2.28 2

1.82 1.77 1

2.32 2.24 2

1.79 1.75 1

2.26 2.19 2

1.77 1.72 1

2.23 2.15 2

1.76 1.71 1

2.20 2.12 2

1.74 1.69 1

2.17 2.09 1

1.72 1.67 1

2.12 2.04 1

1.70 1.65 1

2.09 2.01 1

1.69 1.64 1

2.07 1.99 1

Table A14 of Statistical Methods, 7th 00. by George W. Snedecor and Will iam G. Cochran. Copyright © 1980 by the Iowa State University Pres

2121 South State Avenue, Ames, Iowa 50010. Reprinted with permission of the Iowa State University Press.



734 APPEND IX B STAT IST ICAL TABLES

TABLE B .5 TH E STU DE NTIZE D R AN GE STA TISTIC (q)*

*The critical values for q corresponding to ex= .05 (lightface type) and ex= .01 (boldface type).

dffm:

ErrorTenn 2 3 4 5 6

k =Number of Treatments

8 910 11 12

120

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

24

30

40

60

00

3.64

5.70

3.46

5.24

3.34

4.95

3.26

4.75

3.20

4.60

3.15

4.483.11

4.39

3.08

4.32

3.06

4.26

3.03

4.21

3.01

4.17

3.00

4.13

2.98

4.10

2.97

4.07

2.96

4.05

2.95

4.02

2.92

3.96

2.89

3.89

2.863.82

2.83

3.76

2.80

3.70

2.77

3.64

4.60

6.98

4.34

6.33

4.16

5.92

4.04

5.64

3.95

5.43

3.88

5.273.82

5.15

3.77

5.05

3.73

4.96

3.70

4.89

3.67

4.84

3.65

4.79

3.63

4.74

3.61

4.70

3.59

4.67

3.58

4.64

3.53

4.55

3.49

4.45

3.444.37

3.40

4.28

3.36

4.20

3.31

4.12

5.22

7.80

4.90

7.03

4.68

6.54

4.53

6.20

4.41

5.96

4.33

5.774.26

5.62

4.20

5.50

4.15

5.40

4.11

5.32

4.08

5.25

4.05

5.19

4.02

5.14

4.00

5.09

3.98

5.05

3.96

5.02

3.90

4.91

3.85

4.80

3.794.70

3.74

4.59

3.68

4.50

3.63

4.40

5.67

8.42

5.30

7.56

5.06

7.01

4.89

6.62

4.76

6.35

4.65

6.144.57

5.97

4.51

5.84

4.45

5.73

4.41

5.63

4.37

5.56

4.33

5.49

4.30

5.43

4.28

5.38

4.25

5.33

4.23

5.29

4.17

5.17

4.10

5.05

4.044.93

3.98

4.82

3.92

4.71

3.86

4.60

6.03

8.91

5.63

7.97

5.36

7.37

5.17

6.96

5.02

6.66

4.91

6.434.82

6.25

4.75

6.10

4.69

5.98

4.64

5.88

4.59

5.80

4.56

5.72

4.52

5.66

4.49

5.60

4.47

5.55

4.45

5.51

4.37

5.37

4.30

5.24

4.235.11

4.16

4.99

4.10

4.87

4.03

4.76

6.33

9.32

5.90

8.32

5.61

7.68

5.40

7.24

5.24

6.91

5.12

6.675.03

6.48

4.95

6.32

4.88

6.19

4.83

6.08

4.78

5.99

4.74

5.92

4.70

5.85

4.67

5.79

4.65

5.73

4.62

5.69

4.54

5.54

4.46

5.40

4.395.26

4.31

5.13

4.24

5.01

4.17

4.88

6.58

9.67

6.12

8.61

5.82

7.94

5.60

7.47

5.43

7.13

5.30

6.875.20

6.67

5.12

6.51

5.05

6.37

4.99

6.26

4.94

6.16

4.90

6.08

4.86

6.01

4.82

5.94

4.79

5.89

4.77

5.84

4.68

5.69

4.60

5.54

4.525.39

4.44

5.25

4.36

5.12

4.28

4.99

6.80

9.97

6.32

8.87

6.00

8.17

5.77

7.68

5.59

7.33

5.46

7.055.35

6.84

5.27

6.67

5.19

6.53

5.13

6.41

5.08

6.31

5.03

6.22

4.99

6.15

4.96

6.08

4.92

6.02

4.90

5.97

4.81

5.81

4.72

5.65

4.635.50

4.55

5.36

4.47

5.21

4.39

5.08

6.99

10.24

6.49

9.10

6.16

8.37

5.92

7.86

5.74

7.49

5.60

7.215.49

6.99

5.39

6.81

5.32

6.67

5.25

6.54

5.20

6.44

5.15

6.35

5.11

6.27

5.07

6.20

5.04

6.14

5.01

6.09

4.92

5.92

4.82

5.76

4.735.60

4.65

5.45

4.56

5.30

4.47

5.16

7.17

10.48

6.65

9.30

6.30

8.55

6.05

8.03

5.87

7.65

5.72

7.365.61

7.13

5.51

6.94

5.43

6.79

5.36

6.66

5.31

6.55

5.26

6.46

5.21

6.38

5.17

6.31

5.14

6.25

5.11

6.19

5.01

6.02

4.92

5.85

4.825.69

4.73

5.53

4.64

5.37

4.55

5.23

7.32

10.70

6.79

9.48

6.43

8.71

6.18

8.18

5.98

7.78

5.83

7.495.71

7.25

5.61

7.06

5.53

6.90

5.46

6.77

5.40

6.66

5.35

6.56

5.31

6.48

5.27

6.41

5.23

6.34

5.20

6.28

5.10

6.11

5.00

5.93

4.905.76

4.81

5.60

4.71

5.44

4.62

5.29

Table 29 of E. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, 2nd ed. New York: Cambridge University Press, 1958. Adapted and

reprinted with permission of the Biometrika trustees.




TABLE B .6 CRITICAL VALUES FOR THE PEARSON CORRELATION*

*To be significant, the sample correlation, r, must be greater than or equal to the critical value in the table.

Level of Significance for

One-Tailed Test

.05 .025 .01 .005


Two-Tailed Test

df= n - 2 .10 .05 .02 .01

.988 .997 .9995 .9999

2 .900 .950 .980 .990

3 .805 .878 .934 .959

4 .729 .811 .882 .917

5 .669 .754 .833 .874

6 .622 .707 .789 .834

7 .582 .666 .750 .798

8 .549 .632 .716 .765

9 .521 .602 .685 .73510 .497 .576 .658 .708

11 .476 .553 .634 .684

12 .458 .532 .612 .661

13 .441 .514 .592 .641

14 .426 .497 .574 .623

15 .412 .482 .558 .606

16 .400 .468 .542 .590

17 .389 .456 .528 .575

18 .378 .444 .516 .561

19 .369 .433 .503 .549

20 .360 .423 .492 .537

21 .352 .413 .482 .526

22 .344 .404 .472 .515

23 .337 .396 .462 .505

24 .330 .388 .453 .496

25 .323 .381 .445 .487

26 .317 .374 .437 .479

27 .311 .367 .430 .471

28 .306 .361 .423 .463

29 .301 .355 .416 .456

30 .296 .349 .409 .449

35 .275 .325 .381 .418

40 .257 .304 .358 .393

45 .243 .288 .338 .372

50 .231 .273 .322 .354

60 .211 .250 .295 .325

70 .195 .232 .274 .30280 .183 .217 .256 .283

90 .173 .205 .242 .267

100 .164 .195 .230 .254

Table VI of R. A. Fisher and F. Yates, Statistical Tables for Biological,

Agricultural and Medical Research, 6th ed. London: Longman Group Ltd.,

1974 (previously published by Oliver and Boyd Ltd., Edinburgh). Copyright

©1963 R A. Fisher and F. Yates. Adapted and reprinted with permission of

Pearson Education Limited.




TABLE B .7 CRITICAL VALUES FOR THE SPEARMAN CORRELATION *

*To be significant, the sample correlation, rs, must be greater than or equal to the critical value in the table.


One-Tailed Test

.05 .025 .01 .005


Two-Tailed Test

n .10 .05 .02 .01

4 1.000

5 0.900 1.000 1.000

6 0.829 0.886 0.943 1.000

7 0.714 0.786 0.893 0.929

8 0.643 0.738 0.833 0.881

9 0.600 0.700 0.783 0.833

10 0.564 0.648 0.745 0.794

11 0.536 0.618 0.709 0.755

12 0.503 0.587 0.671 0.727

13 0.484 0.560 0.648 0.703

14 0.464 0.538 0.622 0.675

15 0.443 0.521 0.604 0.654

16 0.429 0.503 0.582 0.635

17 0.414 0.485 0.566 0.615

18 0.401 0.472 0.550 0.600

19 0.391 0.460 0.535 0.584

20 0.380 0.447 0.520 0.570

21 0.370 0.435 0.508 0.556

22 0.361 0.425 0.496 0.544

23 0.353 0.415 0.486 0.532

24 0.344 0.406 0.476 0.521

25 0.337 0.398 0.466 0.511

26 0.331 0.390 0.457 0.501

27 0.324 0.382 0.448 0.491

28 0.317 0.375 0.440 0.483

29 0.312 0.368 0.433 0.475

30 0.306 0.362 0.425 0.467

35 0.283 0.335 0.394 0.433

40 0.264 0.313 0.368 0.405

45 0.248 0.294 0.347 0.382

50 0.235 0.279 0.329 0.363

60 0.214 0.255 0.300 0.331

70 0.190 0.235 0.278 0.307

80 0.185 0.220 0.260 0.287

90 0.174 0.207 0.245 0.271

100 0.165 0.197 0.233 0.257

Zar, J. H. (1972). Signif icance test ing of the Spearman rank correlation

coefficient. Journal of the American Statistical Association, 67,578.

Reprinted with permission from the Journal of the American Statistical

Association. Copyright © 1972 by the American Statistical Association. All

rights reserved.




TABLE B .8 THE CHI-SQUARE DISTRIBUTION*

*The table entries are critical values of X 2•

Critical

x'

Proportion inCritical Region

df 0.10 0.05 0.025 0.01 0.005

1 2.71 3.84 5.02 6.63 7.88

2 4.61 5.99 7.38 9.21 10.60

3 6.25 7.81 9.35 11.34 12.84

4 7.78 9.49 11.14 13.28 14.86

5 9.24 11.07 12.83 15.09 16.75

6 10.64 12.59 14.45 16.81 18.55

7 12.02 14.07 16.01 18.48 20.28

8 13.36 15.51 17.53 20.09 21.96

9 14.68 16.92 19.02 21.67 23.59

10 15.99 18.31 20.48 23.21 25.19

11 17.28 19.68 21.92 24.72 26.76

12 18.55 21.03 23.34 26.22 28.30

13 19.81 22.36 24.74 27.69 29.82

14 21.06 23.68 26.12 29.14 31.32

15 22.31 25.00 27.49 30.58 32.80

16 23.54 26.30 28.85 32.00 34.27

17 24.77 27.59 30.19 33.41 35.72

18 25.99 28.87 31.53 34.81 37.16

19 27.20 30.14 32.85 36.19 38.5820 28.41 31.41 34.17 37.57 40.00

21 29.62 32.67 35.48 38.93 41.40

22 30.81 33.92 36.78 40.29 42.80

23 32.01 35.17 38.08 41.64 44.18

24 33.20 36.42 39.36 42.98 45.56

25 34.38 37.65 40.65 44.31 46.93

26 35.56 38.89 41.92 45.64 48.29

27 36.74 40.11 43.19 46.96 49.64

28 37.92 41.34 44.46 48.28 50.99

29 39.09 42.56 45.72 49.59 52.34

30 40.26 43.77 46.98 50.89 53.67

40 51.81 55.76 59.34 63.69 66.77

50 63.17 67.50 71.42 76.15 79.49

60 74.40 79.08 83.30 88.38 91.95

70 85.53 90.53 95.02 100.42 104.22

80 96.58 101.88 106.63 112.33 116.32

90 107.56 113.14 118.14 124.12 128.30

100 118.50 124.34 129.56 135.81 140.17

Table 8 of E. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, 3rd ed. New York:

Cambridge University Press, 1966. Adapted and reprinted with permission of the Biometrika

trustees.



738 APPENDIX B STATISTICAL TABLES

TABLE B.9A CRITICAL VALUES OF THE MANN-WHITNEY U FOR a = .05*

*Critical values are provided for a one-tailed test at u = .05 (lightface type) and for a two-tailed test at u = .05

(boldface type). To be significant for any given nA and nB, the obtained U must be equal to or less than the critical

value in the table. Dashes (-) in the body of the table indicate that no decision is possible at the stated level of

significance and values of nA and nB.

2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19 0

20 0

o

o

o 0 2

o 1

o 1 2 3

o 1 2

1 2 4 5o 1 2 3

2 3 5 7

1 2 3 5

2 4 6 8

1 3 5 6

3 5 8 10

2 4 6 8

3 6 9 12

2 4 7 10

4 7 11 14

3 5 8 11

5 8 12 16

3 6 9 13

5 9 13 17

4 7 11 14

6 10 15 19

4 8 12 16

7 11 16 21

5 9 13 17

7 12 18 23

5 10 14 19

8 14 19 25

6 11 15 21

o o 2 2 2

1

7

5

11

9

1613

21

17

26

22

31

26

36

31

41

36

46

40

51

45

56

50

61

55

66

59

71

64

3

1

7

5

12

10

1814

23

19

28

24

33

29

39

34

44

39

50

44

55

49

61

54

66

59

72

64

77

70

o 0

3

1

8

6

14

11

1915

25

21

30

26

36

31

42

37

48

42

54

47

60

53

65

59

71

64

77

70

83

75

3 4 4 4

2 2 2 2

9 9 10 11

6 7 7 8

15 16 17 18

11 12 13 13

20 22 23 2517 18 19 20

26 28 30 32

22 24 25 27

33 35 37 39

28 30 32 34

39 41 44 47

34 36 38 41

45 48 51 54

39 42 45 48

51 55 58 62

45 48 52 55

57 61 65 69

51 55 58 62

64 68 72 77

57 61 65 69

79 75 80 84

63 67 72 76

77 82 87 92

67 74 78 83

83 88 94 100

75 80 85 90

89 95 101 107

81 86 92 98

o

o

1

o

1

o

1

o

o

2

1

2

1

2

1

3

1

3

1

3 9 15 20 26

2 6 11 17 22

4 9 16 22 28

2 7 12 18 24

4 10 17 23 30

2 7 13 19 25

4 11 18 25 32

2 8 13 20 27

o 0 0 011

2334556

1223344

4 5 6 7 8 9 10

3445678

6 8 9 11 12 13 155 6 7 8 9 11 12

8 10 12 14 16 17 19

6 8 10 11 13 14 16

11 13 15 17 19 21 24

8 10 12 14 16 18 20

13 15 18 20 23 26 28

10 13 15 17 19 22 24

15 18 21 24 27 30 33

12 15 17 20 23 26 28

17 20 24 27 31 34 37

14 17 20 23 26 29 33

19 23 27 31 34 38 42

16 19 23 26 30 33 37

21 26 30 34 38 42 47

18 22 26 29 33 37 41

24 28 33 37 42 47 51

20 24 28 33 37 41 45

26 31 36 41 46 51 56

22 26 31 36 40 45 50

28 33 39 44 50 55 61

24 29 34 39 44 49 54

30 36 42 48 54 60 65

26 31 37 42 47 53 59

33 39 45 51 57 64 70

28 34 39 45 51 57 63

35 41 48 55 61 68 75

30 36 42 48 55 61 67

37 44 51 58 65 72 80

32 38 45 52 58 65 72

39 47 54 62 69 77 84

34 41 48 55 62 69 76

77 83 89 96 102 109 115

67 75 81 87 93 99 105

82 88 95 102 109 116 123

74 80 86 93 99 106 112

87 94 101 109 116 123 130

78 85 92 99 106 113 119

92 100 107 115 123 130 138

83 90 98 105 112 119 127



TABLE B.9B

19 2

CRITICAL VALUES OF THE MANN-WHITNEY U FOR a = .01*

APPENDIX B STATISTICALTABLES

*Critical values are provided for a one-tailed test at u = .01 (lightface type) and for a two-tailed test at u =

(boldface type). To be significant for any given nA and nB. the obtained U must be equal to or less than the critical va

in the table. Dashes (-) in the body of the table indicate that no decision is possible at the stated level of significan

and values of nA and nB.

2

2

o

o

o

o

o

o

o

1

o

3 4 5

o

1

o

2

1

3

1

4

2

5

3

6

4

6 8 9 10 11 12

2

1

5

3

8

6

11

9

14

12

17

15

21

18

24

21

13

o

2

1

5

3

9

7

12

10

16

13

20

17

23

20

27

24

31

27

35

31

39

34

43

38

47

42

51

45

55

49

59

53

63

56

67

60

14

o

2

1

6

4

10

7

13

11

17

15

22

18

26

22

30

26

34

30

38

34

43

38

47

42

51

46

56

50

60

54

65

58

69

63

73

67

15

o

3

2

7

5

11

8

15

12

19

16

24

20

28

24

33

29

37

33

42

37

47

42

51

46

56

51

61

55

66

60

70

64

75

69

80

73

44 47 50

39 42 45 4

49 53 56

44 47 51

55 59 63

49 53 56

60 65 69 7

54 58 63

66 70 75

60 64 69 7

71 76 82

65 70 74 7

77 82 88

70 75 81

82 88 94 1

75 81 87

88 94 101 1

81 87 93

93 100 107 1

86 92 99 1

16 17 18

1

o

4

3

9

7

15

12

20

17

26

22 2

32

28 3

38 4

33 3

44 4

39 4

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

o

o

1

o

1

o

2

1

3

1

3

2

o

1

o

o

1

o

2

1

2

1

2

1

3

2

3

2

4

2

42

4

3

5

3

4 7

2 5

5 8

3 6

5 9

3 7

6 10

4 7

7 11

5 8

7 12

5 9

8 13

6 10

9 14

6 11

9 15

7 12

10 16

8 13

7

o o 1

o

3

1

5

3

7

5

9

7

11

9

14

11

16

13

1

o

2

1

3

2

4

3

6

4

7

5

8

6

9

7

11

9

12

10

13

11

15

12

16

13

18

15

19

16

20

17

22

18

1

o

3

1

4

3

6

4

7

6

9

7

11

9

2

1

4

2

6

4

7

6

9

7

11

9

13

11

12 15 18 22

10 13 16 18

14 17 21 24

12 15 18 21

16 20 23 27

13 17 20 24

17 22 26 30

15 18 22 26

19 24 28 33

16 20 24 29

21 26 31 36

18 22 27 31

23 28 33 38

19 24 29 34

24 30 36 41

21 26 31 37

26 32 38 44

22 28 33 39

28 34 40 47

24 30 36 42

1

o

3

2

6

4

8

6

11

9

13

11

16

13

19

16

1

o

4

2

7

5

9

7

12

10

15

13

18

16

22

18

25 28

21 24

28 31

24 27

31 35

27 31

34 38

30 34

37 42

33 37

41 46

36 41

44 49

39 44

47 53

42 47

50 56

45 51

53 60

48 54

o o 0

3

2

7

5

12

9

16

13

21

18

26

22

31

27

36

31

41

36

46

41

51

45

56

50

61

55

66

60

71

65

76

70

82

74

87

79

4 4

2 2

8 9

6 6

13 14

10 11

18 19

15 16

23 24

19 21

28 30

24 26

33 36

29 31

38 41

34 37




TABLE B.10 CRITICAL VALUES OF T FOR THE WILCOXON SIGNED-RANKS TEST*

*To be significant, the obtained T must be equal to or less than the critical value. Dashes (-) in the columns indicate

that no decision is possible for the stated exand n.

L e v e l o f S i g n if i c a n ce o r

O n e - T a i l e d T e s t

. 0 5 . 0 2 5 . 0 1 . 0 0 5

L e v e l o f S i g n i fi c a n c eo r

T w o - T a i l ed Te s t

n . 1 0 . 0 5 . 0 2 . 0 1

5 0 - - -6 2 0 - -7 3 2 0 -8 5 3 1 0

9 8 5 3 1

1 0 1 0 8 5 3

1 1 1 3 1 0 7 5

1 2 1 7 1 3 9 7

1 3 2 1 1 7 1 2 9

1 4 2 5 2 1 1 5 1 2

1 5 3 0 2 5 1 9 1 5

1 6 3 5 2 9 2 3 1 9

1 7 4 1 3 4 2 7 2 3

1 8 4 7 4 0 3 2 2 7

1 9 5 3 4 6 3 7 3 2

2 0 6 0 5 2 4 3 3 7

2 1 6 7 5 8 4 9 4 2

2 2 7 5 6 5 5 5 4 8

2 3 8 3 7 3 6 2 5 4

2 4 9 1 8 1 6 9 6 1

2 5 1 0 0 8 9 7 6 6 8

2 6 1 1 0 9 8 8 4 7 5

2 7 1 1 9 1 0 7 9 2 8 3

L e v e l o f S i g n i fi c a n c eo r

O n e - T a i l e d T e s t

. 0 5 . 0 2 5 . 0 1 . 0 0 5

L e v e l o f S i g n if i c a nc e o r

T w o - T ai l e d T e s t

n . 1 0 . 0 5 . 0 2 . 0 1

2 8 1 3 0 1 1 6 1 0 1 9 1

2 9 1 4 0 1 2 6 1 1 0 1 0 0

3 0 1 5 1 1 3 7 1 2 0 1 0 9

3 1 1 6 3 1 4 7 1 3 0 1 1 8

3 2 1 7 5 1 5 9 1 4 0 1 2 8

3 3 1 8 7 1 7 0 1 5 1 1 3 8

3 4 2 0 0 1 8 2 1 6 2 1 4 8

3 5 2 1 3 1 9 5 1 7 3 1 5 9

3 6 2 2 7 2 0 8 1 8 5 1 7 1

3 7 2 4 1 2 2 1 1 9 8 1 8 2

3 8 2 5 6 2 3 5 2 1 1 1 9 4

3 9 2 7 1 2 4 9 2 2 4 2 0 7

4 0 2 8 6 2 6 4 2 3 8 2 2 0

4 1 3 0 2 2 7 9 2 5 2 2 3 3

4 2 3 1 9 2 9 4 2 6 6 2 4 7

4 3 3 3 6 3 1 0 2 8 1 2 6 1

4 4 3 5 3 3 2 7 2 9 6 2 7 6

4 5 3 7 1 3 4 3 3 1 2 2 9 1

4 6 3 8 9 3 6 1 3 2 8 3 0 7

4 7 4 0 7 3 7 8 3 4 5 3 2 2

4 8 4 2 6 3 9 6 3 6 2 3 3 9

4 9 4 4 6 4 1 5 3 7 9 3 5 5

5 0 4 6 6 4 3 4 3 9 7 3 7 3

A d a p t ed f r o m F. W i l c ox o n , S . K. Katti, a n d R. A. W i l c o x , Critical Values and Probability Levels of the

Wilcoxon Rank-Swn Test and the Wilcoxon Signed-Ranks Test. W a y n e , N J : A m e r i c a n C y a n a m i d C o m p a n y ,

1 9 6 3 . A d a p t e d and r e p r in t e dw i t h p e r m i s s i o no f t h e A m e r ic a n C y a n am i d C o m p a ny .

Documents

Appendix B - Statistical Tables