Student's t Table (Critical Values)

How to use these tables

Student's t Distribution

Student's t distribution with ν degrees of freedom has density function:

\( f(t,\nu) = \dfrac{\Gamma\!\left(\tfrac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\;\Gamma\!\left(\tfrac{\nu}{2}\right)} \left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}} \)

As ν → ∞ the distribution converges to the standard Normal N(0,1).

Critical values table

Shows the critical values t for each degrees of freedom ν. The Test type selector decides how the columns are labelled:

• Two tails (default): each column is α = P(|T| > t), the area split between both ends. This is what you need for a two-sided test or a confidence interval. Example: α = 0.05, ν = 20 → t = 2.086.
• One tail: each column is α = P(T > t), the right-tail area, for one-sided tests. The same 2.086 appears in column α = 0.025, because α (two tails) = 2 × α (one tail).
• When ν = ∞ the values match the quantiles of the standard Normal.

Cumulative distribution table

Each cell shows P(T ≤ t) for the selected number of degrees of freedom. To compute two-tailed p-values: p = 2 · P(T > |t_obs|) = 2 · (1 − P(T ≤ |t_obs|)).

Related tables and calculators

• Binomial table
• Student's t distribution calculator
• Standard Normal table
• Hypothesis test for one mean
• Hypothesis test for two means
• CI for the mean (unknown σ)

Frequently asked questions

• How do you read the Student's t table? Choose whether the test is two-tailed (two-sided) or one-tailed (one-sided), then find the row for the degrees of freedom ν and the column for the significance level α: the cell is the critical value t. Example: ν = 10 and α = 0.05 give t = 2.228 with two tails and t = 1.812 with one tail.
• When do you use the t table instead of the normal table? With small samples and unknown population variance you use the Student's t table; as the degrees of freedom grow, the t approaches the standard normal table.
• What is the difference between one tail and two tails? One tail leaves the whole area α on one side (P(T > t) = α); two tails split it between both ends (P(|T| > t) = α), which makes the critical value larger.