Normality and Goodness of Fit
Anderson-Darling Test Table: Critical Values of A²
Critical values of the Anderson-Darling statistic, which gives extra weight to the tails of the distribution.
- Normality and specified distributions
- n = 4 to 100
- Test your data
Anderson-Darling Table
Two tables: normality with estimated mean and SD, and a fully specified distribution. Cells: critical A². Reject if A² ≥ the cell.
Testing normality, mean and SD estimated from the data
| α → n ↓ | 0.15 | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|---|---|---|---|---|---|
| 4 | 0.4348 | 0.4831 | 0.5599 | 0.6297 | 0.6979 | 0.7343 |
| 5 | 0.4638 | 0.5157 | 0.5998 | 0.6851 | 0.7968 | 0.8718 |
| 6 | 0.4806 | 0.5371 | 0.6326 | 0.7257 | 0.8445 | 0.9343 |
| 7 | 0.4931 | 0.5525 | 0.6524 | 0.7510 | 0.8808 | 0.9764 |
| 8 | 0.5025 | 0.5638 | 0.6675 | 0.7706 | 0.9052 | 1.0074 |
| 9 | 0.5096 | 0.5722 | 0.6785 | 0.7841 | 0.9229 | 1.0274 |
| 10 | 0.5154 | 0.5792 | 0.6871 | 0.7947 | 0.9364 | 1.0436 |
| 11 | 0.5196 | 0.5840 | 0.6936 | 0.8032 | 0.9476 | 1.0574 |
| 12 | 0.5234 | 0.5886 | 0.6993 | 0.8095 | 0.9554 | 1.0659 |
| 13 | 0.5265 | 0.5922 | 0.7038 | 0.8156 | 0.9630 | 1.0743 |
| 14 | 0.5289 | 0.5952 | 0.7075 | 0.8198 | 0.9688 | 1.0804 |
| 15 | 0.5311 | 0.5977 | 0.7107 | 0.8235 | 0.9733 | 1.0865 |
| 16 | 0.5333 | 0.6003 | 0.7141 | 0.8281 | 0.9791 | 1.0937 |
| 17 | 0.5348 | 0.6019 | 0.7160 | 0.8302 | 0.9809 | 1.0965 |
| 18 | 0.5366 | 0.6038 | 0.7186 | 0.8329 | 0.9845 | 1.0992 |
| 19 | 0.5378 | 0.6054 | 0.7203 | 0.8352 | 0.9877 | 1.1040 |
| 20 | 0.5390 | 0.6065 | 0.7220 | 0.8376 | 0.9902 | 1.1058 |
| 21 | 0.5400 | 0.6079 | 0.7237 | 0.8394 | 0.9930 | 1.1104 |
| 22 | 0.5409 | 0.6089 | 0.7247 | 0.8409 | 0.9944 | 1.1111 |
| 23 | 0.5420 | 0.6102 | 0.7263 | 0.8426 | 0.9965 | 1.1137 |
| 24 | 0.5426 | 0.6109 | 0.7271 | 0.8437 | 0.9981 | 1.1163 |
| 25 | 0.5433 | 0.6118 | 0.7281 | 0.8445 | 1.0002 | 1.1189 |
| 26 | 0.5439 | 0.6124 | 0.7289 | 0.8460 | 1.0013 | 1.1183 |
| 27 | 0.5443 | 0.6130 | 0.7300 | 0.8468 | 1.0025 | 1.1206 |
| 28 | 0.5451 | 0.6139 | 0.7311 | 0.8481 | 1.0037 | 1.1212 |
| 29 | 0.5461 | 0.6149 | 0.7321 | 0.8493 | 1.0053 | 1.1238 |
| 30 | 0.5463 | 0.6151 | 0.7321 | 0.8494 | 1.0055 | 1.1243 |
| 35 | 0.5481 | 0.6171 | 0.7348 | 0.8529 | 1.0099 | 1.1308 |
| 40 | 0.5497 | 0.6192 | 0.7374 | 0.8558 | 1.0127 | 1.1330 |
| 45 | 0.5508 | 0.6202 | 0.7387 | 0.8575 | 1.0153 | 1.1359 |
| 50 | 0.5518 | 0.6216 | 0.7406 | 0.8596 | 1.0173 | 1.1382 |
| 60 | 0.5531 | 0.6228 | 0.7418 | 0.8616 | 1.0206 | 1.1419 |
| 70 | 0.5541 | 0.6241 | 0.7436 | 0.8636 | 1.0227 | 1.1440 |
| 80 | 0.5548 | 0.6249 | 0.7445 | 0.8646 | 1.0242 | 1.1460 |
| 90 | 0.5552 | 0.6253 | 0.7447 | 0.8650 | 1.0245 | 1.1453 |
| 100 | 0.5558 | 0.6260 | 0.7459 | 0.8660 | 1.0257 | 1.1460 |
Fully specified distribution (no parameters estimated)
| α → n ↓ | 0.15 | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|---|---|---|---|---|---|
| 4 | 1.6316 | 1.9594 | 2.5443 | 3.1560 | 3.9872 | 4.6330 |
| 5 | 1.6285 | 1.9527 | 2.5336 | 3.1400 | 3.9643 | 4.5970 |
| 6 | 1.6272 | 1.9491 | 2.5260 | 3.1260 | 3.9443 | 4.5796 |
| 7 | 1.6259 | 1.9469 | 2.5212 | 3.1191 | 3.9343 | 4.5662 |
| 8 | 1.6256 | 1.9446 | 2.5173 | 3.1154 | 3.9357 | 4.5695 |
| 9 | 1.6252 | 1.9443 | 2.5156 | 3.1133 | 3.9257 | 4.5507 |
| 10 | 1.6252 | 1.9424 | 2.5103 | 3.1050 | 3.9160 | 4.5504 |
| 11 | 1.6247 | 1.9414 | 2.5104 | 3.1038 | 3.9136 | 4.5393 |
| 12 | 1.6245 | 1.9405 | 2.5078 | 3.1012 | 3.9135 | 4.5387 |
| 13 | 1.6233 | 1.9398 | 2.5082 | 3.1012 | 3.9049 | 4.5314 |
| 14 | 1.6232 | 1.9395 | 2.5062 | 3.0965 | 3.9092 | 4.5361 |
| 15 | 1.6228 | 1.9380 | 2.5044 | 3.0980 | 3.9062 | 4.5331 |
| 16 | 1.6234 | 1.9388 | 2.5060 | 3.0982 | 3.9070 | 4.5305 |
| 17 | 1.6230 | 1.9384 | 2.5057 | 3.0966 | 3.9057 | 4.5290 |
| 18 | 1.6237 | 1.9386 | 2.5039 | 3.0935 | 3.9000 | 4.5212 |
| 19 | 1.6235 | 1.9382 | 2.5031 | 3.0933 | 3.9002 | 4.5268 |
| 20 | 1.6229 | 1.9371 | 2.5014 | 3.0935 | 3.9009 | 4.5254 |
| 21 | 1.6228 | 1.9373 | 2.5008 | 3.0909 | 3.9014 | 4.5229 |
| 22 | 1.6227 | 1.9375 | 2.5017 | 3.0911 | 3.8954 | 4.5185 |
| 23 | 1.6224 | 1.9371 | 2.5011 | 3.0921 | 3.9008 | 4.5273 |
| 24 | 1.6228 | 1.9371 | 2.5018 | 3.0941 | 3.9016 | 4.5219 |
| 25 | 1.6229 | 1.9370 | 2.5013 | 3.0894 | 3.8933 | 4.5146 |
| 26 | 1.6225 | 1.9360 | 2.4994 | 3.0891 | 3.8965 | 4.5244 |
| 27 | 1.6228 | 1.9365 | 2.5005 | 3.0871 | 3.8908 | 4.5155 |
| 28 | 1.6223 | 1.9346 | 2.4986 | 3.0869 | 3.8892 | 4.5187 |
| 29 | 1.6224 | 1.9370 | 2.5009 | 3.0901 | 3.8930 | 4.5181 |
| 30 | 1.6235 | 1.9374 | 2.5009 | 3.0900 | 3.8956 | 4.5180 |
| 35 | 1.6227 | 1.9360 | 2.4979 | 3.0841 | 3.8894 | 4.5189 |
| 40 | 1.6216 | 1.9343 | 2.4960 | 3.0849 | 3.8886 | 4.5117 |
| 45 | 1.6214 | 1.9345 | 2.4949 | 3.0823 | 3.8859 | 4.5093 |
| 50 | 1.6210 | 1.9334 | 2.4940 | 3.0807 | 3.8815 | 4.5046 |
| 60 | 1.6211 | 1.9339 | 2.4944 | 3.0810 | 3.8858 | 4.5118 |
| 70 | 1.6217 | 1.9352 | 2.4959 | 3.0848 | 3.8864 | 4.5034 |
| 80 | 1.6226 | 1.9346 | 2.4966 | 3.0825 | 3.8912 | 4.5159 |
| 90 | 1.6208 | 1.9333 | 2.4955 | 3.0820 | 3.8825 | 4.5019 |
| 100 | 1.6217 | 1.9351 | 2.4951 | 3.0809 | 3.8807 | 4.5005 |
Tip: click any value to highlight its row and column.
How to read this table
- Sort the data and compute zᵢ = F(x₍ᵢ₎), using the normal CDF with the sample mean and SD (or the fully specified distribution).
- Compute A² = −n − (1/n) Σ (2i − 1)[ln zᵢ + ln(1 − z₍ₙ₊₁₋ᵢ₎)].
- Use the first table when the mean and SD were estimated from the data, and the second when the distribution was specified in advance.
- Reject if A² is greater than or equal to the table value.
Worked example
A normality check on 20 residuals gives A² = 0.80. Row 20, α = 0.05 of the first table gives 0.7220 and the α = 0.025 value is 0.8376, so 0.025 < p < 0.05.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Critical values are percentiles of A² from 10 million simulated samples for each n. The fully specified case converges to Anderson and Darling's asymptotic values (1.933, 2.492 and 3.857 at α = 0.10, 0.05 and 0.01); the normal case converges to Stephens' values (0.631, 0.752 and 1.035).
Frequently asked questions
What is Stephens' A*?
A small-sample adjustment that lets one set of asymptotic critical values (0.752 at α = 0.05) be used for any n. This table gives exact percentiles of the unadjusted A² for each n instead.
Why use Anderson-Darling?
It weights the tails more heavily than the Kolmogorov-Smirnov statistic, so it is better at detecting heavy tails and outliers.