Hypothesis Tests

Chi-Square Variance Test

Tests whether the population variance equals a hypothesized value (one-sample variance test).

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > 1 Variance

Description

The chi-square variance test evaluates the null hypothesis that the variance of a normally distributed population equals a specified value σ₀². The test statistic χ² = (n−1)·s²/σ₀² follows a chi-squared distribution with n−1 degrees of freedom. The test requires normally distributed data — for non-normal data, a bootstrap approach should be used.

Formulas

χ²

χ^{2} = \frac{( n - 1 ) \cdot s ^{2}}{σ _{0}^{2}}

Test statistic: scaled sample variance under H₀

H₀

H_{0} : σ^{2} = σ_{0}^{2}

Null hypothesis: population variance equals the target value

[\frac{( n - 1 ) s ^{2}}{χ _{1 - α /2}^{2}}, \frac{( n - 1 ) s ^{2}}{χ _{α /2}^{2}}]

Confidence interval for the population variance

Assumptions

Data are normally distributed
Observations are independent
Continuous data

Limitations

Sensitive to departures from normality
For non-normal data, prefer Levene test or bootstrap methods

References

Montgomery, D. C. (2019), Introduction to Statistical Quality Control, 8th Ed., Chapter 4
NIST/SEMATECH e-Handbook of Statistical Methods, Section 7.2.3.2