Which statistics table should I use?

Start from the question. Comparing a mean uses the z table when σ is known and the t table when it is estimated. Comparing three or more means uses the F table, followed by the Tukey table for pairs. Counts in categories use the chi-square table, a correlation uses the Pearson r critical values, and probabilities of counts use the binomial or Poisson tables. The chooser at the top of this page does this for you.

What is the difference between a one-tailed and a two-tailed table?

They are the same numbers with different labels. A two-tailed α splits the error between both tails, so the two-tailed 0.05 column equals the one-tailed 0.025 column. For z that is 1.960. Chi-square and Tukey q tests use the right tail only, and so does the ANOVA F test. For a two-sided test of two variances, use α/2.

How do I find the degrees of freedom?

It depends on the test: n − 1 for one-sample and paired t tests, n1 + n2 − 2 for two independent groups with equal variances, n − 2 for a correlation or a regression slope, k − 1 and N − k for one-way ANOVA, and (rows − 1)(columns − 1) for a chi-square test of independence. The decision table on this page lists them all.

What if my df is not in the table?

Use the next smaller df that is listed, which is slightly conservative, or enter the exact df in the calculator of the linked table page. Software always gives the exact value.

Do I still need tables if software reports p-values?

For reported results, use the software output. Tables remain useful for exams that allow them, for checking a hand calculation, for sample size planning and for understanding which distribution a test is built on.

How are the numbers on this page calculated?

Critical values and probabilities in the chooser are computed in your browser from the exact distribution functions, using the same code as the individual table pages, then rounded to three decimals (four for probabilities). The Tukey q values come from a precomputed grid and are interpolated in 1/df between listed rows. Use statistical software for anything you will publish.

Statistics Tables (Which Table to Use)

Pick the right statistics table in seconds: z, t, chi-square, F, Tukey, binomial and Poisson. Get your df and critical value, then open the full table.

What do you want to do?

Draw it

Shade the rejection region on a bell curve

Open the bell curve generator

Is the population σ known?

Design

Sample size n

Significance level α

Test type

Use this table

T table

Open the full table

Statistic: t statistic
df rule: df = n − 1
Your df: df = 25 − 1 = 24

Critical value t*

2.064

Reject H0 if |t| > 2.064

Two-tailed: use it for "different from". α = 0.05 is split, 0.025 in each tail, and you compare |t| with the value.

Comparing two proportions with a z test uses the Z table instead (no df).

Also see: P-value table (turn your t into p)

Draw it

Shade the rejection region on a bell curve

Open the bell curve generator

All statistics tables by topic

Every table page has its own calculator and printable sheets.

Decision table: test, statistic, table and df

Each example is computed at α = 0.05 (two-tailed for z, t and r).

Statistical tests with the statistic, table, degrees of freedom formula, common alpha levels and an example critical value
Test	Statistic	Table	Degrees of freedom	Common α	Example
One-sample z test (σ known)	z	Z table	none	0.05, 0.01	z* = 1.960
One-sample t test	t	T table	n − 1	0.05, 0.01	n = 25, df = 24: t* = 2.064
Paired t test	t	T table	pairs − 1	0.05, 0.01	25 pairs, df = 24: t* = 2.064
Two independent means (equal variances)	t	T table	n1 + n2 − 2	0.05, 0.01	15 + 15, df = 28: t* = 2.048
Two independent means (Welch)	t	T table	Welch–Satterthwaite	0.05, 0.01	df is fractional: round down or use software
Slope of a simple regression	t	T table	n − 2	0.05, 0.01	n = 22, df = 20: t* = 2.086
Pearson correlation	r (or t)	Critical values table, r	n − 2	0.05, 0.01	n = 12, df = 10: r* = 0.576
One-way ANOVA	F	F distribution table	k − 1 and N − k	0.05, 0.01	k = 3, N = 30: F* = 3.354
Compare two variances	F	F distribution table	n1 − 1 and n2 − 1	0.05 two-sided (use 0.025 column)	15 and 15, two-sided (α/2 = 0.025): F* = 2.979
Tukey post hoc after ANOVA	q	Tukey table	k and N − k	0.05, 0.01	k = 3, df = 27: q* = 3.506
Chi-square goodness of fit	χ²	Chi-square table	categories − 1 − estimated parameters	0.05	4 categories, df = 3: χ²* = 7.815
Chi-square test of independence	χ²	Chi-square table	(rows − 1)(columns − 1)	0.05	3 × 4 table, df = 6: χ²* = 12.592
Successes in n trials	X	Binomial distribution table	none (n and p)	none	n = 10, p = 0.3: P(X ≤ 3) from the table
Events per interval	X	Poisson distribution table	none (λ)	none	λ = 2.5: P(X ≤ 3) from the table

Free Statistics Table Flowcharts and Cheat Sheets

Every sheet below is drawn by the same code as the chooser above, with values computed from the exact distributions. View, download and print them free, no sign-up.

Flowchart

Which Statistics Table Do I Need? Flowchart

Six questions, one table each, with the df rule written under every answer.

1 / 5

Choose a table

Reference tables

Study aids

What are statistics tables?

A statistics table lists critical values or probabilities of a probability distribution, so you can judge a test result without software. Each test statistic follows its own distribution: z follows the standard normal curve, t follows the t distribution, chi-square and F follow skewed right-tailed distributions, and counts follow the binomial or Poisson distribution. The question is therefore not which table is best, but which distribution your statistic comes from. The chooser above answers it from your goal and the few inputs that change the answer, then computes the degrees of freedom and the critical value for you.

How to read any statistics table in three steps

Find the row: work out the degrees of freedom (df) from the rule for your test, for example n − 1 for a one-sample t test or (rows − 1)(columns − 1) for a chi-square test of independence.
Find the column: pick the significance level α, usually 0.05, and for z, t and r decide between a one-tailed and a two-tailed test.
Compare: reject the null hypothesis when the absolute value of your statistic is larger than the table value. For chi-square, Tukey q and the ANOVA F test the test is right-tailed, so the statistic itself is compared.

One-tailed or two-tailed?

Use a two-tailed test unless you fixed one direction before collecting data. The same number can carry two labels: a one-tailed α of 0.05 and a two-tailed α of 0.10 share a column. For t with df = 10 the one-tailed 0.05 value is 1.812, while the two-tailed 0.05 value is 2.228. For z the two-tailed 0.05 value is 1.960 and the two-tailed 0.01 value is 2.576. Chi-square and Tukey q tests use the right tail only, and so does the ANOVA F test. For a two-sided test of two variances, use α/2.

Examples of values you will look up

A t test with 13 observations compared with 0.05 two-tailed uses df = 12 and the value 2.179. A chi-square test with df = 3 at α = 0.05 uses 7.815. A one-way ANOVA with 3 groups and 30 observations has df = (2, 27), and the F table gives 3.354 at α = 0.05. The chooser above reproduces each of these numbers when you enter the matching inputs, and the linked table pages show the neighbouring rows and columns.

What if my degrees of freedom are not listed?

Printed tables skip some rows, for example t tables often jump from 30 to 40. Take the next smaller df that is listed. That gives a slightly larger critical value, so the decision is conservative: you will not call a result significant when it is borderline in the other direction. Where the page offers a calculator, type the exact df instead. A Welch t test has a fractional df that software reports, and the usual rule is to round it down for a printed table.

When to skip the table and use software

Research

Binomial Distribution Table

Binomial distribution table for n = 1 to 20 and 25 with individual and cumulative values, and a calculator up to n = 1000.

Try it free

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Statistics Tables (Which Table to Use)

All statistics tables by topic

Normal and tMeans and standardized scores.

Variances and ANOVASpread, group differences and pairwise follow-ups.

CorrelationIs a relationship real, and how strong is it?

Discrete distributionsCounts instead of measurements.

CombinedSeveral distributions on one page.