Chi-Square Table

What Is a Chi-Square Table?

A chi-square (χ²) table displays critical values for the chi-square distribution. This skewed, right-tailed distribution is used in hypothesis tests involving categorical data and variance analysis.

The chi-square distribution arises when you sum squared standard normal random variables. With k degrees of freedom:

$\chi^2_k = \sum_{i=1}^{k} Z_i^2$

When to Use Chi-Square Tests

The chi-square distribution is used for:

• Goodness of Fit Tests — Does observed data fit an expected distribution?
• Tests of Independence — Are two categorical variables related?
• Homogeneity Tests — Do different populations have the same distribution?
• Variance Tests — Does a population variance equal a hypothesized value?

How to Read the Chi-Square Table

The table shows critical values for the right tail probability. Each row corresponds to a degrees of freedom value, and each column to a significance level (area to the right).

To find χ² for a goodness-of-fit test with 5 categories at α = 0.05:

1. Calculate df = k - 1 = 4
2. Find row df = 4
3. Find column α = 0.05
4. Critical value is χ² = 9.488

Reject H₀ if your test statistic exceeds this critical value.

Degrees of Freedom

Common Applications

Chi-Square Test of Independence

Tests whether two categorical variables are associated:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

Where O = observed frequency, E = expected frequency.

Example: Testing if voting preference is independent of gender using a 2×3 contingency table:

• df = (2 - 1)(3 - 1) = 2
• At α = 0.05, critical χ² = 5.991

Goodness of Fit Test

Tests if observed frequencies match expected probabilities:

Example: Are rolls of a die fair? (6 categories)

• df = 6 - 1 = 5
• At α = 0.05, critical χ² = 11.070

Confidence Intervals for Variance

To construct a CI for population variance σ²:

$\frac{(n-1)s^2}{\chi^2_{\alpha/2}} < \sigma^2 < \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}$

Key Properties

• Always positive: χ² values are ≥ 0
• Right-skewed: Especially for small df
• Approaches normal: As df → ∞, chi-square becomes approximately normal
• Mean = df: The expected value equals the degrees of freedom
• Variance = 2×df: The variance is twice the degrees of freedom

Common Critical Values

For quick reference at α = 0.05:

Related Tools

• Chi-Square Test Calculator — Test of independence
• Chi-Square Goodness of Fit — Test distributions
• Chi-Square Distribution Calculator — Find probabilities
• Fisher's Exact Test — For small samples

Summary

The chi-square table is essential for analyzing categorical data and testing variance hypotheses. Unlike symmetric distributions, chi-square is right-skewed, with the critical value representing the point beyond which only α proportion of the distribution lies. Understanding how to read this table correctly—matching degrees of freedom and significance level—is fundamental for goodness-of-fit tests, independence tests, and variance analysis.