The t-distribution, also called Student's t-distribution, is a continuous probability distribution used for inference about a population mean when the population standard deviation is unknown and the sample standard deviation is used in its place. It was introduced by William Sealy Gosset in 1908 under the pen name "Student" while he worked at Guinness Brewery, where he needed reliable inference from small samples of barley.
The distribution is symmetric and bell-shaped around zero, like the standard normal, but has heavier tails that account for the extra uncertainty introduced by estimating the standard deviation from the sample. Its shape is controlled by a single parameter, the degrees of freedomν (typically ν=n−1 for a one-sample test). As ν increases, the t-distribution converges to the standard normal N(0,1) — by ν≥30 the two are nearly indistinguishable for most practical purposes.
The table above is interactive — pick the variant you need and set your degrees of freedom, and the table regenerates with exact probabilities.
Step 1 — Pick a table variant from the dropdown:
• Cumulative (less than t), Positive Values — gives P(T≤t) for t from 0 to 4.1 in steps of 0.1 • Cumulative (less than t), Negative Values — gives P(T≤t) for t from −4.0 to 0 • Complementary Cumulative (greater than t), Positive Values — gives P(T>t) for t from 0 to 4.1, the right-tail area • Complementary Cumulative (greater than t), Negative Values — gives P(T>t) for t from −4.1 to 0
Step 2 — Enter degrees of freedom in the Degrees of Freedom input (default 30). Each row of the table is one t value at the increment 0.1, evaluated at your ν.
Step 3 — Read off the probability. For a one-tailed test, use the complementary-cumulative variant directly: the value at the table t is your p-value. For a two-tailed test, double it. For a confidence interval, take the cumulative-less-than-t variant and read off the t value at the target probability (e.g., 0.975 for a 95% interval).
The t-distribution has a small number of clean properties that make it tractable for hand calculation and easy to compare to the normal:
• Symmetric around zero — the mean is 0 for ν>1 (undefined for ν=1, the Cauchy case) • Variance — ν−2ν for ν>2, otherwise undefined or infinite; always greater than 1, reflecting the heavier tails • Skewness — 0 for ν>3 • Excess kurtosis — ν−46 for ν>4, positive for all finite ν, decaying to 0 as ν→∞
The symmetry around zero means that for any positive t, P(T≤−t)=P(T≥t) — which is why the negative-value tables in the widget above are mirror images of the positive ones.
As ν→∞, Tν→Z∼N(0,1). At ν=30 the difference is around 1% in the tail; by ν=100 the t and z critical values agree to three decimals.
The t-distribution is the workhorse of small-sample inference about means. The standard scenarios:
• One-sample t-test — testing whether a population mean μ equals a hypothesized value μ0 using t=s/nxˉ−μ0 with ν=n−1 • Two-sample t-test (independent) — comparing two group means; degrees of freedom depend on whether variances are pooled (equal-variance assumption) or unequal (Welch's t-test, with the Satterthwaite df approximation) • Paired t-test — comparing before/after measurements on the same subjects, applied to the differences with ν=n−1 • Confidence interval for a mean — xˉ±tν,α/2⋅ns when σ is unknown • Regression — the standard t-statistic for each coefficient β^j uses a t-distribution with ν=n−p where p is the number of fitted parameters
For all of these, the t-distribution replaces the standard normal whenever the population standard deviation must be estimated from the sample. With large samples (n≥30 as a rough guide) the two are close enough that the z-distribution is often used instead — see the normal distribution table.
Degrees of freedom (ν, or df) measure how many independent observations are available for estimating variability after the parameters of interest have been fitted. For a one-sample test on n observations, ν=n−1 because one degree is consumed by estimating the mean. For two independent samples with equal variances, ν=n1+n2−2; for Welch's unequal-variance test, ν is given by the Satterthwaite approximation. In regression with n observations and p fitted coefficients (including the intercept), ν=n−p.
The critical t-valuetν,α is the value such that P(Tν>tν,α)=α. Lower ν produces a wider distribution and therefore larger critical values, reflecting the extra uncertainty from a small sample. For example, the two-tailed 5% critical value drops from t4,0.025=2.776 to t29,0.025=2.045 to t∞,0.025=1.960 (the normal value).
Use the widget above to read off any tν,α directly. To get the critical value for a given right-tail area α, choose the Complementary Cumulative (greater than t), Positive Values table at your ν and find the row where the probability equals α.