Chi-Square (χ²) Table

What is the Chi-Square Distribution?

The chi-square (χ²) distribution is a right-skewed distribution used in hypothesis testing when working with categorical data or variance. It arises when you sum the squares of independent standard normal variables. The shape is determined by one parameter: degrees of freedom (df).

• Goodness-of-fit test: χ² = Σ (O − E)² / E, where O is observed and E is expected frequency.
• Test of independence: used on contingency tables with df = (rows − 1)(cols − 1).
• Variance test: χ² = (n − 1)s² / σ₀², with df = n − 1.
• As df increases, the distribution becomes more symmetric and approaches a normal shape.

Critical Chi-Square Values (Right-Tailed)

Each cell shows the critical value χ²_{α, df} such that P(χ² > value) = α. Reject H₀ when your test statistic exceeds the critical value for your chosen significance level.

How to Use This Table

• Calculate your test statistic: use χ² = Σ (O − E)² / E for goodness-of-fit, or χ² = (n − 1)s² / σ₀² for variance tests.
• Find your degrees of freedom: for a goodness-of-fit test, df = k − 1 (categories minus 1). For a contingency table, df = (rows − 1)(cols − 1).
• Choose your significance level α and locate the corresponding column.
• Compare: if your χ² statistic exceeds the table value, reject H₀.
• Large df: for df > 100, use the approximation χ² ≈ ½(z + √(2·df − 1))², where z is the normal critical value.

Common Uses of the Chi-Square Test

References

• Khan Academy — Chi-square tests for categorical data.
• OpenStax — Introductory Statistics, Chapter 11: The Chi-Square Distribution.
• Wikipedia — Chi-squared distribution (definition, properties, and table).

See also