Critical Value Tables

Percentages Under Normal Curve

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2518 0.2549

0.7 0.2580 0.2612 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852

0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133

0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389

1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621

1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830

1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015

1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177

1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319

1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441

1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545

1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633

1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706

1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767

2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817

Percentages Under Normal Curve (continued)

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857

2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890

2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916

2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936

2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952

2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964

2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974

2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981

2.9 0.4891 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986

3.0 0.4986 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993

3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995

3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997

3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998 0.4998

3.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998

3.6 0.4998 0.4998 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999

3.7 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999

3.8 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.5000 0.5000 0.5000

3.9 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000

Pearson r (p. < .05 Level)

Level of Significance for Two-Tailed Test

df .10 .05 .02 .01 .002 .001

1 0.988 0.997 1.000 1.000 1.000 1.000

2 0.900 0.950 0.980 0.990 0.998 0.999

3 0.805 0.878 0.934 0.959 0.986 0.991

4 0.729 0.811 0.882 0.917 0.963 0.974

5 0.669 0.754 0.833 0.875 0.935 0.951

6 0.621 0.707 0.789 0.834 0.905 0.925

7 0.582 0.666 0.750 0.798 0.875 0.898

8 0.549 0.632 0.715 0.765 0.847 0.872

9 0.521 0.602 0.685 0.735 0.820 0.847

10 0.497 0.576 0.658 0.708 0.795 0.823

11 0.476 0.553 0.634 0.684 0.772 0.801

12 0.458 0.532 0.612 0.661 0.750 0.780

13 0.441 0.514 0.592 0.641 0.730 0.760

14 0.426 0.497 0.574 0.623 0.711 0.742

15 0.412 0.482 0.558 0.606 0.694 0.725

16 0.400 0.468 0.543 0.590 0.678 0.708

17 0.389 0.456 0.529 0.575 0.662 0.693

18 0.378 0.444 0.516 0.561 0.648 0.679

19 0.369 0.433 0.503 0.549 0.635 0.665

20 0.360 0.423 0.492 0.537 0.622 0.652

Pearson r (continued)

df .10 .05 .02 .01 .002 .001

21 0.352 0.413 0.482 0.526 0.610 0.640

22 0.344 0.404 0.472 0.515 0.599 0.629

23 0.337 0.396 0.462 0.505 0.588 0.618

24 0.330 0.388 0.453 0.496 0.578 0.607

25 0.323 0.381 0.445 0.487 0.568 0.597

26 0.317 0.374 0.437 0.479 0.559 0.588

27 0.311 0.367 0.430 0.471 0.550 0.579

28 0.306 0.361 0.423 0.463 0.541 0.570

30 0.296 0.349 0.409 0.449 0.526 0.554

40 0.257 0.304 0.358 0.393 0.463 0.490

50 0.231 0.273 0.322 0.354 0.419 0.443

60 0.211 0.250 0.295 0.325 0.385 0.408

80 0.183 0.217 0.257 0.283 0.336 0.357

100 0.164 0.195 0.230 0.254 0.303 0.321

200 0.116 0.138 0.164 0.181 0.216 0.230

500 0.073 0.088 0.104 0.115 0.138 0.146

Critical Values for t (p. < .05 Level)

Level of Significance for Two-Tailed Test

df .10 .05 .02 .01 .002 .001

1 6.314 12.706 31.820 63.657 318.309 636.619

2 2.920 4.303 6.965 9.925 22.327 31.599

3 2.353 3.182 4.541 5.841 10.215 12.924

4 2.132 2.776 3.747 4.604 7.173 8.610

5 2.015 2.571 3.365 4.032 5.893 6.869

6 1.943 2.447 3.143 3.707 5.208 5.959

7 1.895 2.365 2.998 3.499 4.785 5.408

8 1.860 2.306 2.896 3.355 4.501 5.041

9 1.833 2.262 2.821 3.250 4.297 4.781

10 1.812 2.228 2.764 3.169 4.144 4.587

11 1.796 2.201 2.718 3.106 4.025 4.437

12 1.782 2.179 2.681 3.055 3.930 4.318

13 1.771 2.160 2.650 3.012 3.852 4.221

14 1.761 2.145 2.624 2.977 3.787 4.140

15 1.753 2.131 2.602 2.947 3.733 4.073

16 1.746 2.120 2.583 2.921 3.686 4.015

17 1.740 2.110 2.567 2.898 3.646 3.965

18 1.734 2.101 2.552 2.878 3.610 3.922

19 1.729 2.093 2.539 2.861 3.579 3.883

20 1.725 2.086 2.528 2.845 3.552 3.850

Critical values of t (continued)

df .10 .05 .02 .01 .002 .001

21 1.721 2.080 2.518 2.831 3.527 3.819

22 1.717 2.074 2.508 2.819 3.505 3.792

23 1.714 2.069 2.500 2.807 3.485 3.768

24 1.711 2.064 2.492 2.797 3.467 3.745

25 1.708 2.060 2.485 2.787 3.450 3.725

26 1.706 2.056 2.479 2.779 3.435 3.707

27 1.703 2.052 2.473 2.771 3.421 3.690

28 1.701 2.048 2.467 2.763 3.408 3.674

29 1.699 2.045 2.462 2.756 3.396 3.659

30 1.697 2.042 2.457 2.750 3.385 3.646

50 1.676 2.009 2.403 2.678 3.261 3.496

100 1.660 1.984 2.364 2.626 3.174 3.390

150 & over 1.645 1.960 2.326 2.576 3.090 3.291

Chi-Square (p. < .05 Level)

Level of Significance for Two-Tailed Test

df .10 .05 .02 .01 .002 .001

1 2.706 3.841 5.412 6.635 9.550 10.828

2 4.605 5.991 7.824 9.210 12.430 13.816

3 6.251 7.815 9.837 11.345 14.796 16.267

4 7.779 9.488 11.668 13.277 16.924 18.468

5 9.236 11.071 13.388 15.086 18.908 20.516

6 10.645 12.592 15.033 16.812 20.792 22.459

7 12.017 14.067 16.622 18.475 22.601 24.323

8 13.362 15.507 18.168 20.090 24.353 26.124

9 14.684 16.919 19.679 21.666 26.056 27.877

10 15.987 18.307 21.161 23.209 27.722 29.588

11 17.275 19.675 22.618 24.725 29.354 31.264

12 18.549 21.026 24.054 26.217 30.957 32.909

13 19.812 22.362 25.472 27.688 32.535 34.528

14 21.064 23.685 26.873 29.141 34.091 36.123

15 22.307 24.996 28.259 30.578 35.628 37.697

16 23.542 26.296 29.633 32.000 37.146 39.252

17 24.769 27.587 30.995 33.409 38.648 40.790

Chi Square (Continued)

Level of Significance for Two-Tailed Test

df .10 .05 .02 .01 .002 .001

18 25.989 28.869 32.346 34.805 40.136 42.312

19 27.204 30.144 33.687 36.191 41.610 43.820

20 28.412 31.410 35.020 37.566 43.072 45.315

21 29.615 32.671 36.343 38.932 44.522 46.797

22 30.813 33.924 37.659 40.289 45.962 48.268

23 32.007 35.172 38.968 41.638 47.391 49.728

24 33.196 36.415 40.270 42.980 48.812 51.179

25 34.382 37.652 41.566 44.314 50.223 52.620

26 35.563 38.885 42.856 45.642 51.627 54.052

27 36.741 40.113 44.140 46.963 53.023 55.476

28 37.916 41.337 45.419 48.278 54.411 56.892

29 39.087 42.557 46.693 49.588 55.792 58.301

30 40.256 43.773 47.962 50.892 57.167 59.703

50 63.167 67.505 72.613 76.154 83.657 86.661

100 118.498 124.342 131.142 135.807 145.577 149.449

500 540.930 553.127 567.070 576.493 595.882 603.446

Distribution of F (p. < .05 Level)

Between

1 2 3 4 5 6 8 12 24 µ

Within

1 161.40 199.50 215.70 224.60 230.20 234.00 238.90 243.90 249.00 254.30

2 18.51 19.00 19.16 19.25 19.30 19.33 19.37 19.41 19.45 19.50

3 10.13 9.55 9.28 9.12 9.01 8.94 8.84 8.74 8.64 8.53

4 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.91 5.77 5.63

5 6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.68 4.53 4.36

6 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.00 3.84 3.67

7 5.59 4.74 4.35 4.12 3.97 3.87 3.73 3.57 3.41 3.23

8 5.32 4.46 4.07 3.84 3.69 3.58 3.44 3.28 3.12 2.93

9 5.12 4.26 3.86 3.63 3.48 3.37 3.23 3.07 2.90 2.71

10 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.91 2.74 2.54

11 4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.79 2.61 2.40

12 4.75 3.88 3.49 3.26 3.11 3.00 2.85 2.69 2.50 2.30

13 4.67 3.80 3.41 3.18 3.02 2.92 2.77 2.60 2.42 2.21

14 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.53 2.35 2.13

15 4.54 3.68 3.29 3.06 2.90 2.79 2.64 2.48 2.29 2.07

16 4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.42 2.24 2.01

17 4.45 3.59 3.20 2.96 2.81 2.70 2.55 2.38 2.19 1.96

Distribution of F (Continued)

Between

1 2 3 4 5 6 8 12 24 µ

Within

18 4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.34 2.15 1.92

19 4.38 3.52 3.13 2.90 2.74 2.63 2.48 2.31 2.11 1.88

20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.28 2.08 1.84

21 4.32 3.47 3.07 2.84 2.68 2.57 2.42 2.25 2.05 1.81

22 4.30 3.44 3.05 2.82 2.66 2.55 2.40 2.53 2.03 1.78

23 4.28 3.42 3.03 2.80 2.64 2.53 2.38 2.20 2.00 1.76

24 4.26 3.40 3.01 2.78 2.62 2.51 2.36 2.18 1.98 1.73

25 4.24 3.38 2.99 2.76 2.60 2.49 2.34 2.16 1.96 1.71

26 4.22 3.37 2.98 2.74 2.59 2.47 2.32 2.15 1.95 1.69

27 4.21 3.35 2.96 2.73 2.57 2.46 2.30 2.13 1.93 1.67

28 4.20 3.34 2.95 2.71 2.56 2.44 2.29 2.12 1.91 1.65

29 4.18 3.33 2.93 2.70 2.54 2.43 2.28 2.10 1.90 1.64

30 4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.09 1.89 1.62

40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.00 1.79 1.51

60 4.00 3.45 2.76 2.52 2.37 2.25 2.10 1.92 1.70 1.39

120 3.92 3.07 2.68 2.45 2.29 2.17 2.02 1.83 1.61 1.25

µ 3.84 2.99 2.60 2.37 2.21 2.09 1.94 1.75 1.52 1.00