How to read the table: each cell shows the quantile χ²(p,ν) such that P(X ≤ χ²) = p. For a goodness-of-fit test at the 5% level with ν = 10 degrees of freedom, look up the column p = 0.950 → χ² = 18.307.
Click a cell to see the critical value.
χ²(p, ν) — quantiles for p = P(X ≤ χ²)
How to read the table: each cell shows P(X ≤ x) for the selected degrees of freedom. The row gives the integer part and first decimal; the column adds the second decimal.
Click a cell to see the value.
P(X ≤ x, ν) for x from 0.0 to 30.9
How to read the table: each cell shows the density f(x, ν). The χ² distribution is defined for x ≥ 0. As ν increases, the curve becomes more symmetric and shifts to the right.
Click a cell to see the value.
f(x, ν) for x from 0.0 to 30.9
How to use these tables
Chi-Square Distribution
If Z₁, Z₂, …, Zᵥ are independent standard normal variables, then X = Z₁² + … + Zᵥ² follows a chi-square distribution with ν degrees of freedom. Its density function is:
\( f(x,\nu) = \dfrac{x^{\nu/2-1}\,e^{-x/2}}{2^{\nu/2}\,\Gamma(\nu/2)}, \quad x \geq 0 \)
The mean is ν and the variance is 2ν.
Critical values table
Shows the quantiles χ²(p, ν) such that P(X ≤ χ²) = p. For a test at significance level α (right tail) with ν degrees of freedom, use the column p = 1 − α.
- Example: test at 5%, ν = 5 → p = 0.95 → χ² = 11.070.
- Columns with p < 0.5 correspond to the left tail (small values of χ²).
Main uses
- Goodness-of-fit test
- Test of independence in contingency tables
- Variance estimation and confidence intervals for σ²
- Homogeneity tests