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D.1 Calculating Quantiles with Mathematica

D.1 Calculating Quantiles with Mathematica

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396



Appendix D: Tables of Distribution Quantiles



Table D.1 Integral of the standardized normal distribution (3.10) and (3.12) from 0 to z in steps

of 0.01

z

0

1

2

3

4

5

6

7

8

9

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9



0.0000

0.0398

0.0793

0.1179

0.1554

0.1915

0.2257

0.2580

0.2881

0.3159

0.3413

0.3643

0.3849

0.4032

0.4192

0.4332

0.4452

0.4554

0.4641

0.4713

0.4772

0.4821

0.4861

0.4893

0.4918

0.4938

0.4953

0.4965

0.4974

0.4981

0.4987

0.4990

0.4993

0.4995

0.4997

0.4998

0.4998

0.4999

0.4999

0.5000



0.0040

0.0438

0.0832

0.1217

0.1591

0.1950

0.2291

0.2611

0.2910

0.3186

0.3438

0.3665

0.3869

0.4049

0.4207

0.4345

0.4463

0.4564

0.4649

0.4719

0.4778

0.4826

0.4864

0.4896

0.4920

0.4940

0.4955

0.4966

0.4975

0.4982

0.4987

0.4991

0.4993

0.4995

0.4997

0.4998

0.4998

0.4999

0.4999

0.5000



0.0080

0.0478

0.0871

0.1255

0.1628

0.1985

0.2324

0.2642

0.2939

0.3212

0.3461

0.3686

0.3888

0.4066

0.4222

0.4357

0.4474

0.4573

0.4656

0.4726

0.4783

0.4830

0.4868

0.4898

0.4922

0.4941

0.4956

0.4967

0.4976

0.4982

0.4987

0.4991

0.4994

0.4995

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0120

0.0517

0.0910

0.1293

0.1664

0.2019

0.2357

0.2673

0.2967

0.3238

0.3485

0.3708

0.3907

0.4082

0.4236

0.4370

0.4484

0.4582

0.4664

0.4732

0.4788

0.4834

0.4871

0.4901

0.4925

0.4943

0.4957

0.4968

0.4977

0.4983

0.4988

0.4991

0.4994

0.4996

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0160

0.0557

0.0948

0.1331

0.1700

0.2054

0.2389

0.2704

0.2995

0.3264

0.3508

0.3729

0.3925

0.4099

0.4251

0.4382

0.4495

0.4591

0.4671

0.4738

0.4793

0.4838

0.4875

0.4904

0.4927

0.4945

0.4959

0.4969

0.4977

0.4984

0.4988

0.4992

0.4994

0.4996

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0199

0.0596

0.0987

0.1368

0.1736

0.2088

0.2422

0.2734

0.3023

0.3289

0.3531

0.3749

0.3944

0.4115

0.4265

0.4394

0.4505

0.4599

0.4678

0.4744

0.4798

0.4842

0.4878

0.4906

0.4929

0.4946

0.4960

0.4970

0.4978

0.4984

0.4989

0.4992

0.4994

0.4996

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0239

0.0636

0.1026

0.1406

0.1772

0.2123

0.2454

0.2764

0.3051

0.3315

0.3554

0.3770

0.3962

0.4131

0.4279

0.4406

0.4515

0.4608

0.4686

0.4750

0.4803

0.4846

0.4881

0.4909

0.4931

0.4948

0.4961

0.4971

0.4979

0.4985

0.4989

0.4992

0.4994

0.4996

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0279

0.0675

0.1064

0.1443

0.1808

0.2157

0.2486

0.2794

0.3078

0.3340

0.3577

0.3790

0.3980

0.4147

0.4292

0.4418

0.4525

0.4616

0.4693

0.4756

0.4808

0.4850

0.4884

0.4911

0.4932

0.4949

0.4962

0.4972

0.4979

0.4985

0.4989

0.4992

0.4995

0.4996

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0319

0.0714

0.1103

0.1480

0.1844

0.2190

0.2517

0.2823

0.3106

0.3365

0.3599

0.3810

0.3997

0.4162

0.4306

0.4429

0.4535

0.4625

0.4699

0.4761

0.4812

0.4854

0.4887

0.4913

0.4934

0.4951

0.4963

0.4973

0.4980

0.4986

0.4990

0.4993

0.4995

0.4996

0.4997

0.4998

0.4999

0.4999

0.4999

0.5000



0.0359

0.0754

0.1141

0.1517

0.1879

0.2224

0.2549

0.2852

0.3133

0.3389

0.3621

0.3830

0.4015

0.4177

0.4319

0.4441

0.4545

0.4633

0.4706

0.4767

0.4817

0.4857

0.4890

0.4916

0.4936

0.4952

0.4964

0.4974

0.4981

0.4986

0.4990

0.4993

0.4995

0.4997

0.4998

0.4998

0.4999

0.4999

0.4999

0.5000



Appendix D: Tables of Distribution Quantiles



397



Fig. D.1 [Top left] Definite integral of the standard normal distribution (3.10) and (3.12) from 0 to

z. [Top right] Definition of the pth quantile of the χ2 distribution (3.21). [Bottom left] Definition

of the pth quantile of the t distribution (3.22). [Bottom right] Definition of the pth quantile of the

F distribution (3.23)



0.995th quantile of the Student’s t distribution with ν = 1 degree of freedom and

the 95. percentile of the F distribution for ν1 = ν2 = 10 we issue the commands

Quantile[ChiSquareDistribution[5], 0.90],

Quantile[StudentTDistribution[1], 0.995],

Quantile[FRatioDistribution[10,10], 0.95],



which give (in the same order as above)

9.23636,

63.6567,

2.97824.



(Compare these values to entries in the corresponding Tables.) Definite integrals of

all mentioned distributions can be obtained by commands of the form

NIntegrate[PDF[FRatioDistribution[7,9],

NIntegrate[PDF[FRatioDistribution[7,9],

NIntegrate[PDF[FRatioDistribution[7,9],

NIntegrate[PDF[FRatioDistribution[9,7],



x],

x],

x],

x],



{x,

{x,

{x,

{x,



0,

0,

0,

0,



3.293}],

3.70}],

5.613}],

1./3.70}].



Here we have only demonstrated a sample calculation of integrating the density of the

F distribution with parameters required by the Example on p. 187: the four command

lines listed above yield the values



398



Appendix D: Tables of Distribution Quantiles



Table D.2 Values of the erf function (3.8) from 0 to z in steps of 0.01

z

0

1

2

3

4

5

6

7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0



0.0000

0.1125

0.2227

0.3286

0.4284

0.5205

0.6039

0.6778

0.7421

0.7969

0.8427

0.8802

0.9103

0.9340

0.9523

0.9661

0.9763

0.9838

0.9891

0.9928

0.9953

0.9970

0.9981

0.9989

0.9993

0.9996

0.9998

0.9999

0.9999

1.000

1.000



0.0113

0.1236

0.2335

0.3389

0.4380

0.5292

0.6117

0.6847

0.7480

0.8019

0.8468

0.8835

0.9130

0.9361

0.9539

0.9673

0.9772

0.9844

0.9895

0.9931

0.9955

0.9972

0.9982

0.9989

0.9993

0.9996

0.9998

0.9999

0.9999

1.000

1.000



0.0226

0.1348

0.2443

0.3491

0.4475

0.5379

0.6194

0.6914

0.7538

0.8068

0.8508

0.8868

0.9155

0.9381

0.9554

0.9684

0.9780

0.9850

0.9899

0.9934

0.9957

0.9973

0.9983

0.9990

0.9994

0.9996

0.9998

0.9999

0.9999

1.000

1.000



0.0338

0.1459

0.2550

0.3593

0.4569

0.5465

0.6270

0.6981

0.7595

0.8116

0.8548

0.8900

0.9181

0.9400

0.9569

0.9695

0.9788

0.9856

0.9903

0.9937

0.9959

0.9974

0.9984

0.9990

0.9994

0.9997

0.9998

0.9999

0.9999

1.000

1.000



0.0451

0.1569

0.2657

0.3694

0.4662

0.5549

0.6346

0.7047

0.7651

0.8163

0.8586

0.8931

0.9205

0.9419

0.9583

0.9706

0.9796

0.9861

0.9907

0.9939

0.9961

0.9975

0.9985

0.9991

0.9994

0.9997

0.9998

0.9999

0.9999

1.000

1.000



0.0564

0.1680

0.2763

0.3794

0.4755

0.5633

0.6420

0.7112

0.7707

0.8209

0.8624

0.8961

0.9229

0.9438

0.9597

0.9716

0.9804

0.9867

0.9911

0.9942

0.9963

0.9976

0.9985

0.9991

0.9995

0.9997

0.9998

0.9999

0.9999

1.000

1.000



0.0676

0.1790

0.2869

0.3893

0.4847

0.5716

0.6494

0.7175

0.7761

0.8254

0.8661

0.8991

0.9252

0.9456

0.9611

0.9726

0.9811

0.9872

0.9915

0.9944

0.9964

0.9977

0.9986

0.9992

0.9995

0.9997

0.9998

0.9999

0.9999

1.000

1.000



0.0789

0.1900

0.2974

0.3992

0.4937

0.5798

0.6566

0.7238

0.7814

0.8299

0.8698

0.9020

0.9275

0.9473

0.9624

0.9736

0.9818

0.9877

0.9918

0.9947

0.9966

0.9979

0.9987

0.9992

0.9995

0.9997

0.9998

0.9999

1.000

1.000

1.000



8



9



0.0901

0.2009

0.3079

0.4090

0.5027

0.5879

0.6638

0.7300

0.7867

0.8342

0.8733

0.9048

0.9297

0.9490

0.9637

0.9745

0.9825

0.9882

0.9922

0.9949

0.9967

0.9980

0.9987

0.9992

0.9995

0.9997

0.9998

0.9999

1.000

1.000

1.000



0.1013

0.2118

0.3183

0.4187

0.5117

0.5959

0.6708

0.7361

0.7918

0.8385

0.8768

0.9076

0.9319

0.9507

0.9649

0.9755

0.9832

0.9886

0.9925

0.9951

0.9969

0.9980

0.9988

0.9993

0.9996

0.9998

0.9999

0.9999

1.000

1.000

1.000



0.95001,

0.96385,

0.99001,

0.03615 = 1-0.96385.



The calculation for other distributions proceeds along the same lines.



χ2.005



0.000

0.010

0.072

0.207

0.412

0.676

0.989

1.34

1.73

2.16

2.60

3.07

3.57

4.07

4.60

5.14

5.70

6.26

6.84

7.43

8.03

8.64



ν



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22



0.0002

0.0201

0.115

0.297

0.554

0.872

1.24

1.65

2.09

2.56

3.05

3.57

4.11

4.66

5.23

5.81

6.41

7.01

7.63

8.26

8.90

9.54



χ2.01



0.0010

0.0506

0.216

0.484

0.831

1.24

1.69

2.18

2.70

3.25

3.82

4.40

5.01

5.63

6.26

6.91

7.56

8.23

8.91

9.59

10.3

11.0



χ2.025



0.0039

0.103

0.352

0.711

1.15

1.64

2.17

2.73

3.33

3.94

4.57

5.23

5.89

6.57

7.26

7.96

8.67

9.39

10.1

10.9

11.6

12.3



χ2.05

0.0158

0.211

0.584

1.06

1.61

2.20

2.83

3.49

4.17

4.87

5.58

6.30

7.04

7.79

8.55

9.31

10.1

10.9

11.7

12.4

13.2

14.0



χ2.1

0.102

0.575

1.21

1.92

2.67

3.45

4.25

5.07

5.90

6.74

7.58

8.44

9.30

10.2

11.0

11.9

12.8

13.7

14.6

15.5

16.3

17.2



χ2.25

0.455

1.39

2.37

3.36

4.35

5.35

6.35

7.34

8.34

9.34

10.3

11.3

12.3

13.3

14.3

15.3

16.3

17.3

18.3

19.3

20.3

21.3



χ2.5

1.32

2.77

4.11

5.39

6.63

7.84

9.04

10.2

11.4

12.5

13.7

14.8

16.0

17.1

18.2

19.4

20.5

21.6

22.7

23.8

24.9

26.0



χ2.75

2.71

4.61

6.25

7.78

9.24

10.6

12.0

13.4

14.7

16.0

17.3

18.5

19.8

21.1

22.3

23.5

24.8

26.0

27.2

28.4

29.6

30.8



χ2.90

3.84

5.99

7.81

9.49

11.1

12.6

14.1

15.5

16.9

18.3

19.7

21.0

22.4

23.7

25.0

26.3

27.6

28.9

30.1

31.4

32.7

33.9



χ2.95

5.02

7.38

9.35

11.1

12.8

14.4

16.0

17.5

19.0

20.5

21.9

23.3

24.7

26.1

27.5

28.8

30.2

31.5

32.9

34.2

35.5

36.8



χ2.975

6.63

9.21

11.3

13.3

15.1

16.8

18.5

20.1

21.7

23.2

24.7

26.2

27.7

29.1

30.6

32.0

33.4

34.8

36.2

37.6

38.9

40.3



χ2.99

7.88

10.6

12.8

14.9

16.7

18.5

20.3

22.0

23.6

25.2

26.8

28.3

29.8

31.3

32.8

34.3

35.7

37.2

38.6

40.0

41.4

42.8



χ2.995



(continued)



10.8

13.8

16.3

18.5

20.5

22.5

24.3

26.1

27.9

29.6

31.3

32.9

34.5

36.1

37.7

39.3

40.8

42.3

43.8

45.3

46.8

48.3



χ2.999



Table D.3 Quantiles χ2p of the χ2 distribution (3.21) with ν degrees of freedom for some typical (most commonly used) values of p from 0.005 to 0.999



Appendix D: Tables of Distribution Quantiles

399



χ2.005



9.26

9.89

10.5

11.2

11.8

12.5

13.1

13.8

20.7

28.0

35.5

43.3

51.2

59.2

67.3



ν



23

24

25

26

27

28

29

30

40

50

60

70

80

90

100



10.2

10.9

11.5

12.2

12.9

13.6

14.3

15.0

22.2

29.7

37.5

45.4

53.5

61.8

70.1



χ2.01



Table D.3 (continued)



11.7

12.4

13.1

13.8

14.6

15.3

16.0

16.8

24.4

32.4

40.5

48.8

57.2

65.6

74.2



χ2.025



13.1

13.8

14.6

15.4

16.2

16.9

17.7

18.5

26.5

34.8

43.2

51.7

60.4

69.1

77.9



χ2.05

14.8

15.7

16.5

17.3

18.1

18.9

19.8

20.6

29.1

37.7

46.5

55.3

64.3

73.3

82.4



χ2.1

18.1

19.0

19.9

20.8

21.7

22.7

23.6

24.5

33.7

42.9

52.3

61.7

71.1

80.6

90.1



χ2.25

22.3

23.3

24.3

25.3

26.3

27.3

28.3

29.3

39.3

49.3

59.3

69.3

79.3

89.3

99.3



χ2.5

27.1

28.2

29.3

30.4

31.5

32.6

33.7

34.8

45.6

56.3

67.0

77.6

88.1

98.6

109



χ2.75

32.0

33.2

34.4

35.6

36.7

37.9

39.1

40.3

51.8

63.2

74.4

85.5

96.6

108

118



χ2.90

35.2

36.4

37.7

38.9

40.1

41.3

42.6

43.8

55.8

67.5

79.1

90.5

102

113

124



χ2.95

38.1

39.4

40.6

41.9

43.2

44.5

45.7

47.0

59.3

71.4

83.3

95.0

107

118

130



χ2.975

41.6

43.0

44.3

45.6

47.0

48.3

49.6

50.9

63.7

76.2

88.4

100

112

124

136



χ2.99

44.2

45.6

46.9

48.3

49.6

51.0

52.3

53.7

66.8

79.5

92.0

104

116

128

140



χ2.995

49.7

51.2

52.6

54.1

55.5

56.9

58.3

59.7

73.4

86.7

99.6

112

125

137

149



χ2.999



400

Appendix D: Tables of Distribution Quantiles



t0.55



0.158

0.142

0.137

0.134

0.132

0.131

0.130

0.130

0.129

0.129

0.129

0.128

0.128

0.128

0.128

0.128

0.128

0.127

0.127

0.127

0.127



ν



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21



0.325

0.289

0.277

0.271

0.267

0.265

0.263

0.262

0.261

0.260

0.260

0.259

0.259

0.258

0.258

0.258

0.257

0.257

0.257

0.257

0.257



t0.60



0.727

0.617

0.584

0.569

0.559

0.553

0.549

0.546

0.543

0.542

0.540

0.539

0.538

0.537

0.536

0.535

0.534

0.534

0.533

0.533

0.532



t0.70

1.000

0.816

0.765

0.741

0.727

0.718

0.711

0.706

0.703

0.700

0.697

0.695

0.694

0.692

0.691

0.690

0.689

0.688

0.688

0.687

0.686



t0.75

1.376

1.061

0.978

0.941

0.920

0.906

0.896

0.889

0.883

0.879

0.876

0.873

0.870

0.868

0.866

0.865

0.863

0.862

0.861

0.860

0.859



t0.80

3.08

1.89

1.64

1.53

1.48

1.44

1.41

1.40

1.38

1.37

1.36

1.36

1.35

1.35

1.34

1.34

1.33

1.33

1.33

1.33

1.32



t0.90

6.31

2.92

2.35

2.13

2.02

1.94

1.89

1.86

1.83

1.81

1.80

1.78

1.77

1.76

1.75

1.75

1.74

1.73

1.73

1.72

1.72



t0.95

12.7

4.30

3.18

2.78

2.57

2.45

2.36

2.31

2.26

2.23

2.20

2.18

2.16

2.14

2.13

2.12

2.11

2.10

2.09

2.09

2.08



t0.975

31.8

6.96

4.54

3.75

3.36

3.14

3.00

2.90

2.82

2.76

2.72

2.68

2.65

2.62

2.60

2.58

2.57

2.55

2.54

2.53

2.52



t0.99

63.7

9.92

5.84

4.60

4.03

3.71

3.50

3.36

3.25

3.17

3.11

3.05

3.01

2.98

2.95

2.92

2.90

2.88

2.86

2.85

2.83



t0.995



(continued)



3183

70.7

22.2

13.0

9.68

8.02

7.06

6.44

6.01

5.69

5.45

5.26

5.11

4.99

4.88

4.79

4.71

4.65

4.59

4.54

4.49



t0.999



Table D.4 Quantiles t p of the Student’s t distribution (3.22) with ν degrees of freedom for some typical (most commonly used) values of p from 0.55 to 0.999



Appendix D: Tables of Distribution Quantiles

401



t0.55



0.127

0.127

0.127

0.127

0.127

0.127

0.127

0.127

0.127

0.126

0.126

0.126

0.126



ν



22

23

24

25

26

27

28

29

30

40

60

120





Table D.4 (continued)



t0.60



0.256

0.256

0.256

0.256

0.256

0.256

0.256

0.256

0.256

0.255

0.254

0.254

0.253



t0.70



0.532

0.532

0.531

0.531

0.531

0.531

0.530

0.530

0.530

0.529

0.527

0.526

0.524



t0.75

0.686

0.685

0.685

0.684

0.684

0.684

0.683

0.683

0.683

0.681

0.679

0.677

0.674



t0.80

0.858

0.858

0.857

0.856

0.856

0.855

0.855

0.854

0.854

0.851

0.848

0.845

0.842



t0.90

1.32

1.32

1.32

1.32

1.31

1.31

1.31

1.31

1.31

1.30

1.30

1.29

1.28



t0.95

1.72

1.71

1.71

1.71

1.71

1.70

1.70

1.70

1.70

1.68

1.67

1.66

1.64



2.07

2.07

2.06

2.06

2.06

2.05

2.05

2.05

2.04

2.02

2.00

1.98

1.96



t0.975

2.51

2.50

2.49

2.49

2.48

2.47

2.47

2.46

2.46

2.42

2.39

2.36

2.21



t0.99

2.82

2.81

2.80

2.79

2.78

2.77

2.76

2.76

2.75

2.70

2.66

2.62

2.58



t0.995



t0.999

4.45

4.42

4.38

4.35

4.32

4.30

4.28

4.25

4.23

4.09

3.96

3.84

3.72



402

Appendix D: Tables of Distribution Quantiles



ν2 = 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

22

24

26



161

18.5

10.1

7.71

6.61

5.99

5.59

5.32

5.12

4.96

4.84

4.75

4.67

4.60

4.54

4.49

4.45

4.41

4.38

4.35

4.30

4.26

4.23



200

19.0

9.55

6.94

5.79

5.14

4.74

4.46

4.26

4.10

3.98

3.89

3.81

3.74

3.68

3.63

3.59

3.55

3.52

3.49

3.44

3.40

3.37



ν1 = 1 2



216

19.2

9.28

6.59

5.41

4.76

4.35

4.07

3.86

3.71

3.59

3.49

3.41

3.34

3.29

3.24

3.20

3.16

3.13

3.10

3.05

3.01

2.98



3



225

19.3

9.12

6.39

5.19

4.53

4.12

3.84

3.63

3.48

3.36

3.26

3.18

3.11

3.06

3.01

2.96

2.93

2.90

2.87

2.82

2.78

2.74



4



230

19.3

9.01

6.26

5.05

4.39

3.97

3.69

3.48

3.33

3.20

3.11

3.03

2.96

2.90

2.85

2.81

2.77

2.74

2.71

2.66

2.62

2.59



5

234

19.3

8.94

6.16

4.95

4.28

3.87

3.58

3.37

3.22

3.09

3.00

2.92

2.85

2.79

2.74

2.70

2.66

2.63

2.60

2.55

2.51

2.47



6

237

19.4

8.89

6.09

4.88

4.21

3.79

3.50

3.29

3.14

3.01

2.91

2.83

2.76

2.71

2.66

2.61

2.58

2.54

2.51

2.46

2.42

2.39



7

239

19.4

8.85

6.04

4.82

4.15

3.73

3.44

3.23

3.07

2.95

2.85

2.77

2.70

2.64

2.59

2.55

2.51

2.48

2.45

2.40

2.36

2.32



8

241

19.4

8.81

6.00

4.77

4.10

3.68

3.39

3.18

3.02

2.90

2.80

2.71

2.65

2.59

2.54

2.49

2.46

2.42

2.39

2.34

2.30

2.27



9

242

19.4

8.79

5.96

4.74

4.06

3.64

3.35

3.14

2.98

2.85

2.75

2.67

2.60

2.54

2.49

2.45

2.41

2.38

2.35

2.30

2.25

2.22



10

244

19.4

8.74

5.91

4.68

4.00

3.57

3.28

3.07

2.91

2.79

2.69

2.60

2.53

2.48

2.42

2.38

2.34

2.31

2.28

2.23

2.18

2.15



12

246

19.4

8.70

5.86

4.62

3.94

3.51

3.22

3.01

2.85

2.72

2.62

2.53

2.46

2.40

2.35

2.31

2.27

2.23

2.20

2.15

2.11

2.07



15

248

19.5

8.66

5.80

4.56

3.87

3.44

3.15

2.94

2.77

2.65

2.54

2.46

2.39

2.33

2.28

2.23

2.19

2.16

2.12

2.07

2.03

1.99



20

249

19.5

8.64

5.77

4.53

3.84

3.41

3.12

2.90

2.74

2.61

2.51

2.42

2.35

2.29

2.24

2.19

2.15

2.11

2.08

2.03

1.98

1.95



24

250

19.5

8.62

5.75

4.50

3.81

3.38

3.08

2.86

2.70

2.57

2.47

2.38

2.31

2.25

2.19

2.15

2.11

2.07

2.04

1.98

1.94

1.90



30

251

19.5

8.59

5.72

4.46

3.77

3.34

3.04

2.83

2.66

2.53

2.43

2.34

2.27

2.20

2.15

2.10

2.06

2.03

1.99

1.94

1.89

1.85



40



Table D.5 95. percentiles (F0.95 ) of the F distribution (3.23); ν1 degrees of freedom in the numerator and ν2 in the denominator

60

252

19.5

8.57

5.69

4.43

3.74

3.30

3.01

2.79

2.62

2.49

2.38

2.30

2.22

2.16

2.11

2.06

2.02

1.98

1.95

1.89

1.84

1.80



120





254

19.5

8.53

5.63

4.37

3.67

3.23

2.93

2.71

2.54

2.40

2.30

2.21

2.13

2.07

2.01

1.96

1.92

1.88

1.84

1.78

1.73

1.69



(continued)



253

19.5

8.55

5.66

4.40

3.70

3.27

2.97

2.75

2.58

2.45

2.34

2.25

2.18

2.11

2.06

2.01

1.97

1.93

1.90

1.84

1.79

1.75



Appendix D: Tables of Distribution Quantiles

403



28

30

40

60

120





4.20

4.17

4.08

4.00

3.92

3.84



3.34

3.32

3.23

3.15

3.07

3.00



ν1 = 1 2



Table D.5 (continued)



3



2.95

2.92

2.84

2.76

2.68

2.41



4



2.71

2.69

2.61

2.53

2.45

2.11



5



2.56

2.53

2.45

2.37

2.29

1.92



6

2.45

2.42

2.34

2.25

2.18

1.79



7

2.36

2.33

2.25

2.17

2.09

1.70



8

2.29

2.27

2.18

2.10

2.02

1.62



9

2.24

2.21

2.12

2.04

1.96

1.56



10

2.19

2.16

2.08

1.99

1.91

1.52



12

2.12

2.09

2.00

1.92

1.83

1.44



15

2.04

2.01

1.92

1.84

1.75

1.37



20

1.96

1.93

1.84

1.75

1.66

1.28



24

1.91

1.89

1.79

1.70

1.61

1.52



30

1.87

1.84

1.74

1.65

1.55

1.46



40

1.82

1.79

1.69

1.59

1.50

1.39



60

1.77

1.74

1.64

1.53

1.43

1.32



120

1.71

1.68

1.58

1.47

1.35

1.22





1.65

1.62

1.51

1.39

1.25

1.00



404

Appendix D: Tables of Distribution Quantiles



ν2 = 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

22

24

26



4052

98.5

34.1

21.2

16.3

13.8

12.3

11.3

10.6

10.0

9.65

9.33

9.07

8.86

8.68

8.53

8.40

8.29

8.18

8.10

7.95

7.82

7.72



5000

99.0

30.8

18.0

13.3

10.9

9.55

8.65

8.02

7.56

7.21

6.93

6.70

6.51

6.36

6.23

6.11

6.01

5.93

5.85

5.72

5.61

5.53



ν1 = 1 2



5403

99.2

29.5

16.7

12.1

9.78

8.45

7.59

6.99

6.55

6.22

5.95

5.74

5.56

5.42

5.29

5.18

5.09

5.01

4.94

4.82

4.72

4.64



3



5625

99.3

28.7

16.0

11.4

9.15

7.85

7.01

6.42

5.99

5.67

5.41

5.21

5.04

4.89

4.77

4.67

4.58

4.50

4.43

4.31

4.22

4.14



4



5764

99.3

28.2

15.5

11.0

8.75

7.46

6.63

6.06

5.64

5.32

5.06

4.86

4.69

4.56

4.44

4.34

4.25

4.17

4.10

3.99

3.90

3.82



5

5859

99.3

27.9

15.2

10.7

8.47

7.19

6.37

5.80

5.39

5.07

4.82

4.62

4.46

4.32

4.20

4.10

4.01

3.94

3.87

3.76

3.67

3.59



6

5928

99.4

27.7

15.0

10.5

8.26

6.99

6.18

5.61

5.20

4.89

4.64

4.44

4.28

4.14

4.03

3.93

3.84

3.77

3.70

3.59

3.50

3.42



7

5981

99.4

27.5

14.8

10.3

8.10

6.84

6.03

5.47

5.06

4.74

4.50

4.30

4.14

4.00

3.89

3.79

3.71

3.63

3.56

3.45

3.36

3.29



8

6022

99.4

27.4

14.7

10.2

7.98

6.72

5.91

5.35

4.94

4.63

4.39

4.19

4.03

3.89

3.78

3.68

3.60

3.52

3.46

3.35

3.26

3.18



9

6056

99.4

27.2

14.6

10.1

7.87

6.62

5.81

5.26

4.85

4.54

4.30

4.10

3.94

3.80

3.69

3.59

3.51

3.43

3.37

3.26

3.17

3.09



10

6106

99.4

27.1

14.4

9.89

7.72

6.47

5.67

5.11

4.71

4.40

4.16

3.96

3.80

3.67

3.55

3.46

3.37

3.30

3.23

3.12

3.03

2.96



12

6157

99.4

26.9

14.2

9.72

7.56

6.31

5.52

4.96

4.56

4.25

4.01

3.82

3.66

3.52

3.41

3.31

3.23

3.15

3.09

2.98

2.89

2.81



15

6209

99.5

26.7

14.0

9.55

7.40

6.16

5.36

4.81

4.41

4.10

3.86

3.66

3.51

3.37

3.26

3.16

3.08

3.00

2.94

2.83

2.74

2.66



20

6235

99.5

26.6

13.9

9.47

7.31

6.07

5.28

4.73

4.33

4.02

3.78

3.59

3.43

3.29

3.18

3.08

3.00

2.92

2.86

2.75

2.66

2.58



24

6261

99.5

26.5

13.8

9.38

7.23

5.99

5.20

4.65

4.25

3.94

3.70

3.51

3.35

3.21

3.10

3.00

2.92

2.84

2.78

2.67

2.58

2.50



30

6287

99.5

26.4

13.8

9.29

7.14

5.91

5.12

4.57

4.17

3.86

3.62

3.43

3.27

3.13

3.02

2.92

2.84

2.76

2.69

2.58

2.49

2.42



40



Table D.6 99. percentiles (F0.99 ) of the F distribution (3.23); ν1 degrees of freedom in the numerator and ν2 in the denominator

6313

99.5

26.3

13.7

9.20

7.06

5.82

5.03

4.48

4.08

3.78

3.54

3.34

3.18

3.05

2.93

2.83

2.75

2.67

2.61

2.50

2.40

2.33



60



6366

99.5

26.1

13.5

9.02

6.88

5.65

4.86

4.31

3.91

3.60

3.36

3.17

3.00

2.87

2.75

2.65

2.57

2.49

2.42

2.31

2.21

2.13







(continued)



6339

99.5

26.2

13.6

9.11

6.97

5.74

4.95

4.40

4.00

3.69

3.45

3.25

3.09

2.96

2.84

2.75

2.66

2.58

2.52

2.40

2.31

2.23



120



Appendix D: Tables of Distribution Quantiles

405



28

30

40

60

120





7.64

7.56

7.31

7.08

6.85

6.63



5.45

5.39

5.18

4.98

4.79

4.61



ν1 = 1 2



4.57

4.51

4.31

4.13

3.95

3.78



3



Table D.6 (continued)



4.07

4.02

3.83

3.65

3.48

3.32



4



3.75

3.70

3.51

3.34

3.17

3.02



5

3.53

3.47

3.29

3.12

2.96

2.80



6

3.36

3.30

3.12

2.95

2.79

2.64



7

3.23

3.17

2.99

2.82

2.66

2.51



8

3.12

3.07

2.89

2.72

2.56

2.41



9

3.03

2.98

2.80

2.63

2.47

2.32



10

2.90

2.84

2.66

2.50

2.34

2.18



12

2.75

2.70

2.52

2.35

2.19

2.04



15

2.60

2.55

2.37

2.20

2.03

1.88



20

2.52

2.47

2.29

2.12

1.95

1.79



24

2.44

2.39

2.20

2.03

1.86

1.70



30

2.35

2.30

2.11

1.94

1.76

1.59



40

2.26

2.21

2.02

1.84

1.66

1.47



60

2.17

2.11

1.92

1.73

1.53

1.32



120

2.06

2.01

1.80

1.60

1.38

1.00







406

Appendix D: Tables of Distribution Quantiles



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D.1 Calculating Quantiles with Mathematica

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