Statistical Tables & Calculators
This page replaces the tables at the back of your textbook. Pick a distribution, type your statistic (or your α), and get the exact p-value or critical value, with the tail area drawn so you can see what the number means.
🎛️ Distribution calculator
Everything updates as you type. The shaded region is the probability being reported.
How to use it
- Got a test statistic? Whether it came from software, a homework problem, or a formula, choose its distribution, enter it, and read the exact p-value. For t and z, pick the tail your hypothesis calls for; χ² and F tests are naturally right-tailed.
- Need a critical value? Switch to the α mode. The shaded region is the rejection region: a statistic inside it has p < α.
- Checking a printed table? The "critical values at common α" row below the chart is the entire relevant column of a z, t, χ², or F table for your degrees of freedom.
These are exact values, not approximations. The calculator uses the same special functions (incomplete gamma and beta) that R and Python use under the hood, so the numbers match pnorm, pt, pchisq, and pf to many decimal places.
Reading a printed table
Many statistics exams are still taken on paper, with a calculator that does arithmetic and nothing else. The probabilities come from tables at the back of the book, and reading one is a skill in its own right: find the row, find the column, read the cell. Below are the three tables an introductory course gives you, computed with the same exact functions as the calculator above.
Table A gives the area under the standard normal curve to the left of a z. Rows carry z to one decimal place and columns carry the second, so z = −2.41 is where the −2.4 row meets the .01 column, and the cell reads .0080. Table D holds critical values of t. Rows are degrees of freedom, columns are the area left in the upper tail, and the confidence levels along the bottom are for the matching two-tailed interval. Its last row, df = ∞, is the normal distribution, and that row is where z* comes from. Table F does the same job for χ².
Almost every Table A question is one of three moves. An area to the left is what the table prints, so find the cell and stop. An area to the right is 1 minus the printed value, and because the curve is symmetric you can instead look up −z and read that cell as it stands. An area between two values is a subtraction: take the smaller of the two left-areas away from the larger.
Expect the table and this calculator to disagree in the fourth decimal now and then. A two-tailed p read off Table A as 2 × .0228 = .0456 is .0455 exactly. Both are right. The difference comes from the table rounding before you double, and markers know this.
📄 Printed tables
Table A, Table D and Table F, laid out the way a book prints them. Point at a cell or arrow into it to light up its row and column, and click one to send it back to the calculator. Whatever the calculator currently holds is marked in orange.
Table A · standard normal probabilities
Every cell is the area to the left of that z. Row gives the first decimal, column adds the second.
| Generated in your browser from the same functions the calculator uses. |
Table D · t distribution critical values
Each cell is the t with that much area above it. Confidence levels along the foot are for the matching two-tailed interval.
| Generated in your browser from the same functions the calculator uses. |
Table F · χ² critical values
Each cell is the χ² with that much area above it. Chi-square tests read the upper tail only.
| Generated in your browser from the same functions the calculator uses. |
Tables for the rank tests
The rank-based tests read their tables the other way round. A t or a χ² is significant when it is large, so Table D and Table F print a floor to clear. Mann–Whitney's U and Wilcoxon's W both count something that gets smaller as the groups separate, so their tables print a ceiling: you reject when the statistic is at or below the cell. Spearman's ρs behaves like a correlation again, so its table is a floor.
Each cell below is the most generous cutoff whose exact tail probability is still at or under α. That is the convention printed tables use, and it means a cell is often a little stricter than α. With twelve pairs, for example, no value of W has a tail of exactly .025, so the table takes the last one below it.
📄 Printed tables · rank tests
Mann–Whitney U at three significance levels, Wilcoxon's W, and Spearman's ρs. Point at a cell or arrow into it to light up its row and column; click one and the line above says what you can conclude from it.
Table U · Mann–Whitney critical values, two-tailed α = .05
Take the smaller of U1 and U2 and reject when it is at or below the cell. Rows are n2, columns n1; the table is symmetric, so it does not matter which group you call which. A dash means no split of those two samples can reach this α.
| Counted in your browser from the exact null distribution. |
Table U · Mann–Whitney critical values, two-tailed α = .01
The same table at the stricter level. Much of it is blank, because small samples cannot reach this level however the ranks fall.
| Counted in your browser from the exact null distribution. |
Table U · Mann–Whitney critical values, one-tailed α = .05
Use this one only if the direction was predicted before the data arrived, and check that the smaller U belongs to the group you predicted would score higher.
| Counted in your browser from the exact null distribution. |
Table W · Wilcoxon signed-ranks critical values
Drop every pair whose difference is zero, then let n be the pairs that remain. Rank the sizes of the differences, add the ranks that carried a plus sign and the ranks that carried a minus sign, take the smaller total, and reject when it is at or below the cell.
| Counted in your browser from the exact null distribution. |
Table ρ · Spearman critical values
A floor, not a ceiling: reject when |ρs| reaches the cell. Rows up to n = 13 are exact. Larger n use the Edgeworth series that R uses, which matches the exact value to three decimals at every n that can be checked, except n = 15 in the .01 column, where it gives .657 against an exact .654.
| Counted in your browser from the exact null distribution. |
Practice the lookup
Knowing the moves is not the same as using them under time pressure with a paper table in front of you. This drill has ten questions drawn from all eight tables above. Type what the table gives you. The check accepts the table's rounded value as well as the exact one, because on paper the rounded value is the answer.
🎯 Look it up
No timer, no score kept anywhere. Press Enter to check, then Enter again for the next question.
left = read the cell · right = 1 − read, or read −z · between = larger − smaller · U and W = a ceiling · t, χ² and ρ = a floor
The first question appears here.
Common critical values to memorize
| Confidence / α (two-tailed) | z critical | Where you'll meet it |
|---|---|---|
| 90% · α = .10 | 1.645 | Lenient exploratory cutoffs |
| 95% · α = .05 | 1.960 | The default in most of science |
| 99% · α = .01 | 2.576 | Stricter confirmatory tests |
| 99.9% · α = .001 | 3.291 | Very strong evidence claims |
t critical values are always a little larger than these, because t has heavier tails, and they shrink toward the z values as degrees of freedom grow. By df ≈ 120 the difference barely matters. Try it above.
Where these distributions come from
- Normal (z). The bell curve behind z-scores and the Central Limit Theorem. See the lesson →
- Student's t. Its fat-tailed cousin, used when the SD is estimated from the sample. See the lesson →
- Chi-square (χ²) is the distribution of squared deviations, used for tests on counts. See the lesson →
- F is the ratio of two variances, used in ANOVA and regression tests. See the lesson →
Want to drag their parameters around and watch the shapes respond? Try the Distribution Playground. The formula sheet has every formula these tests are built from.