How to read the table: each cell shows the critical t value such that P(T > t) = α (one-tailed) or P(|T| > t) = α (two-tailed). For a two-tailed test at 5%, look up the column α₂ = 0.05. Example with 10 degrees of freedom: t = 2.228.
Click a cell to read the critical value.
How to read the table: each cell shows P(T ≤ t) for the selected degrees of freedom. For negative t: P(T ≤ −t) = 1 − P(T ≤ t), or switch the sign with the button.
Click a cell to read the value.
P(T ≤ t) for t from 0.00 to 6.09
How to read the table: each cell shows the density f(t, ν). The function is symmetric: f(−t, ν) = f(t, ν). As ν increases the curve approaches the standard Normal.
Click a cell to read the value.
f(t, ν) for t from 0.00 to 6.09 (symmetric: f(−t,ν) = f(t,ν))
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 quantiles tα,ν such that P(T > tα,ν) = α (one-tailed test). For a two-tailed test at level α₂, use column α₁ = α₂/2.
- Example: two-tailed test α = 0.05, ν = 20 → column α₁ = 0.025 → t = 2.086.
- 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 > |tobs|) = 2 · (1 − P(T ≤ |tobs|)).