Chi-Square to P Value Calculator

Convert chi-square statistics to p-values for goodness-of-fit, independence, and variance tests. Supports one‑tailed and two‑tailed tests with interactive chi‑square distribution visualization.

Chi‑square distribution: Used in goodness‑of‑fit, independence (contingency tables), and variance tests. p‑value = probability of observing a χ² as extreme as yours under the null hypothesis.

χ²=3.84, df=1 (α=0.05)
χ²=5.99, df=2 (α=0.05)
χ²=7.81, df=3 (α=0.05)
χ²=9.49, df=4 (α=0.05)
χ²=0.5, df=2 (left tail)
Computing...

Understanding the χ² distribution and p‑value

The chi‑square (χ²) distribution is a continuous probability distribution that arises when summing the squares of independent standard normal variables. It is widely used in hypothesis testing, particularly for categorical data.

Mathematical Definition:

For a chi‑square statistic χ² with k degrees of freedom, the p‑value depends on the alternative hypothesis:

  • Right‑tailed: p = P(χ²k > χ²) — most common (goodness‑of‑fit, independence)
  • Left‑tailed: p = P(χ²k < χ²) — used when testing if a variance is less than expected
  • Two‑tailed: p = 2·min[P(χ² > χ²), P(χ² < χ²)] — for two‑sided variance tests

Note: The χ² distribution is non‑negative and right‑skewed.

Common critical χ² values (right‑tailed, α = 0.05)

df Critical χ² df Critical χ²
1 3.841 6 12.592
2 5.991 7 14.067
3 7.815 8 15.507
4 9.488 9 16.919
5 11.070 10 18.307

Interpretation Guidelines

1

Small p‑value (typically ≤ 0.05): Strong evidence against the null hypothesis. The result is statistically significant.

2

Large p‑value (> 0.05): Weak evidence against the null; fail to reject the null hypothesis.

3

Right‑tailed is default: Most chi‑square tests (goodness‑of‑fit, independence) are right‑tailed because we reject when observed and expected frequencies differ too much, leading to a large χ².

Applications

  • Goodness‑of‑fit test: Whether sample data match a hypothesized distribution.
  • Test of independence: Whether two categorical variables are associated (contingency tables).
  • Test of a single variance: Compare sample variance to a hypothesized value.
  • McNemar's test: Paired categorical data.

Calculator features:

  • Uses jStat library for precise χ² CDF and inverse CDF
  • Interactive graph with shaded tail area corresponding to your test type
  • Displays all three p‑values (right, left, two‑tailed) for complete insight
  • Adjustable p‑value format (decimal or scientific)
  • Shows critical χ² value(s) for your selected test and α, with rejection region description

Frequently Asked Questions

A chi‑square test is any statistical test where the test statistic follows a chi‑square distribution under the null hypothesis. Common uses include testing goodness‑of‑fit, independence in contingency tables, and homogeneity.

In goodness‑of‑fit and independence tests, large discrepancies between observed and expected frequencies produce large χ² values. Therefore, we reject the null only when χ² is sufficiently large (right tail). Small χ² values indicate good fit, so they are not evidence against the null.

χ² is defined as a sum of squared terms, so it is always non‑negative. A value of zero indicates perfect agreement between observed and expected frequencies. Negative values are impossible.

It uses the jStat library, which implements high‑accuracy numerical approximations for the chi‑square distribution. Results match standard chi‑square tables and statistical software (R, SAS, SPSS).

As df increases, the chi‑square distribution approaches a normal distribution. The calculator handles large df (e.g., 1000) accurately.