Critical Values for the Chi-Squared Distribution

df

0.995

0.99

0.975

0.95

0.9

0.1

0.05

0.025

0.01

0.005

1

0.000

0.000

0.001

0.004

0.016

2.706

3.841

5.024

6.635

7.879

2

0.010

0.020

0.051

0.103

0.211

4.605

5.991

7.378

9.210

10.597

3

0.072

0.115

0.216

0.352

0.584

6.251

7.815

9.348

11.345

12.838

4

0.207

0.297

0.484

0.711

1.064

7.779

9.488

11.143

13.277

14.860

5

0.412

0.554

0.831

1.145

1.610

9.236

11.070

12.832

15.086

16.750

6

0.676

0.872

1.237

1.635

2.204

10.645

12.592

14.449

16.812

18.548

7

0.989

1.239

1.690

2.167

2.833

12.017

14.067

16.013

18.475

20.278

8

1.344

1.647

2.180

2.733

3.490

13.362

15.507

17.535

20.090

21.955

9

1.735

2.088

2.700

3.325

4.168

14.684

16.919

19.023

21.666

23.589

10

2.156

2.558

3.247

3.940

4.865

15.987

18.307

20.483

23.209

25.188

11

2.603

3.053

3.816

4.575

5.578

17.275

19.675

21.920

24.725

26.757

12

3.074

3.571

4.404

5.226

6.304

18.549

21.026

23.337

26.217

28.300

13

3.565

4.107

5.009

5.892

7.041

19.812

22.362

24.736

27.688

29.819

14

4.075

4.660

5.629

6.571

7.790

21.064

23.685

26.119

29.141

31.319

15

4.601

5.229

6.262

7.261

8.547

22.307

24.996

27.488

30.578

32.801

16

5.142

5.812

6.908

7.962

9.312

23.542

26.296

28.845

32.000

34.267

17

5.697

6.408

7.564

8.672

10.085

24.769

27.587

30.191

33.409

35.718

18

6.265

7.015

8.231

9.390

10.865

25.989

28.869

31.526

34.805

37.156

19

6.844

7.633

8.907

10.117

11.651

27.204

30.144

32.852

36.191

38.582

20

7.434

8.260

9.591

10.851

12.443

28.412

31.410

34.170

37.566

39.997

21

8.034

8.897

10.283

11.591

13.240

29.615

32.671

35.479

38.932

41.401

22

8.643

9.542

10.982

12.338

14.041

30.813

33.924

36.781

40.289

42.796

23

9.260

10.196

11.689

13.091

14.848

32.007

35.172

38.076

41.638

44.181

24

9.886

10.856

12.401

13.848

15.659

33.196

36.415

39.364

42.980

45.558

25

10.520

11.524

13.120

14.611

16.473

34.382

37.652

40.646

44.314

46.928

26

11.160

12.198

13.844

15.379

17.292

35.563

38.885

41.923

45.642

48.290

27

11.808

12.878

14.573

16.151

18.114

36.741

40.113

43.195

46.963

49.645

28

12.461

13.565

15.308

16.928

18.939

37.916

41.337

44.461

48.278

50.994

29

13.121

14.256

16.047

17.708

19.768

39.087

42.557

45.722

49.588

52.335

30

13.787

14.953

16.791

18.493

20.599

40.256

43.773

46.979

50.892

53.672

31

14.458

15.655

17.539

19.281

21.434

41.422

44.985

48.232

52.191

55.002

32

15.134

16.362

18.291

20.072

22.271

42.585

46.194

49.480

53.486

56.328

33

15.815

17.073

19.047

20.867

23.110

43.745

47.400

50.725

54.775

57.648

34

16.501

17.789

19.806

21.664

23.952

44.903

48.602

51.966

56.061

58.964

35

17.192

18.509

20.569

22.465

24.797

46.059

49.802

53.203

57.342

60.275

36

17.887

19.233

21.336

23.269

25.643

47.212

50.998

54.437

58.619

61.581

37

18.586

19.960

22.106

24.075

26.492

48.363

52.192

55.668

59.893

62.883

38

19.289

20.691

22.878

24.884

27.343

49.513

53.384

56.895

61.162

64.181

39

19.996

21.426

23.654

25.695

28.196

50.660

54.572

58.120

62.428

65.475

40

20.707

22.164

24.433

26.509

29.051

51.805

55.758

59.342

63.691

66.766

50

27.991

29.707

32.357

34.764

37.689

63.167

67.505

71.420

76.154

79.490

60

35.534

37.485

40.482

43.188

46.459

74.397

79.082

83.298

88.379

91.952

70

43.275

45.442

48.758

51.739

55.329

85.527

90.531

95.023

100.425

104.215

80

51.172

53.540

57.153

60.391

64.278

96.578

101.879

106.629

112.329

116.321

90

59.196

61.754

65.647

69.126

73.291

107.565

113.145

118.136

124.116

128.299

100

67.328

70.065

74.222

77.929

82.358

118.498

124.342

129.561

135.807

140.170

150

109.142

112.668

117.985

122.692

128.275

172.581

179.581

185.800

193.207

198.360

200

152.241

156.432

162.728

168.279

174.835

226.021

233.994

241.058

249.445

255.264