Student’s t distribution
The t distribution describes the ratio of an estimate’s error to its estimated standard error when the data are normal and the standard deviation is estimated from the sample. It looks like the normal distribution but with heavier tails, which reflect the extra uncertainty of small samples. Its shape depends on the degrees of freedom (df); as df grows, it approaches the standard normal distribution.
What the calculator gives
- the p-value of a t statistic, two-tailed, right-tailed or left-tailed;
- the critical t for a chosen significance level, as in a printed t table;
- the cumulative probability P(T ≤ t) and the density at t;
- the matching normal critical value, for comparison.
Example
A test gives t = 2.1 with 15 degrees of freedom:
- two-tailed p ≈ 0.053: just above 0.05;
- critical t at α = 0.05, two-tailed: 2.131;
- the normal distribution would give 1.96, showing how much wider the t distribution is with few degrees of freedom.
Common critical values (two-tailed, α = 0.05)
| df | Critical t |
|---|---|
| 1 | 12.706 |
| 2 | 4.303 |
| 5 | 2.571 |
| 10 | 2.228 |
| 20 | 2.086 |
| 30 | 2.042 |
| 60 | 2.000 |
| ∞ | 1.960 |
Where the degrees of freedom come from
- one-sample or paired t-test: n − 1;
- two-sample t-test with pooled variance: n₁ + n₂ − 2; with Welch’s correction, a fractional value from the Welch–Satterthwaite formula;
- regression coefficient: n − p − 1, where p is the number of predictors;
- Pearson correlation test: n − 2.
Formulas
The density is
f(t) = Γ((ν + 1)/2) ÷ (√(νπ) Γ(ν/2)) × (1 + t²/ν)^−(ν+1)/2
The calculator computes the cumulative probability with the regularized incomplete beta function, accurate to many digits for any degrees of freedom, including fractional ones.
Tips
- Decide on one or two tails before looking at the data.
- With more than about 100 degrees of freedom, t and normal critical values differ by less than 1%.
- A p-value is not the probability that the null hypothesis is true; it is the probability of data at least this extreme if it were.