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Chi-Square Test for Standard Deviation

NIST/SEMATECH Section 1.3.5.8 Chi-Square Test for Standard Deviation

What It Is

The chi-square test for the standard deviation tests whether a population standard deviation (or variance) equals a specified value. It uses the chi-square distribution to assess whether the observed sample variance is consistent with the hypothesized population variance.

When to Use It

Use this test when a process or product specification defines an acceptable level of variability and you need to verify whether the observed variability meets that standard. It is commonly applied in manufacturing quality control, where specifications may require the process standard deviation to be below a target value. This is a one-sample test for variability, analogous to the one-sample t-test for means.

How to Interpret

Compare the computed chi-square statistic to the critical values from the chi-square distribution with n-1 degrees of freedom. For a two-sided test, reject H0 if the statistic falls outside the interval defined by the lower and upper critical values. For a one-sided test against excessive variability (the more common case), reject if the statistic exceeds the upper critical value. The confidence interval for the variance provides a range of plausible values for the population variance. Note that the chi-square distribution is asymmetric, so the confidence interval for the variance is also asymmetric.

Assumptions and Limitations

This test requires the data to be drawn from a normal distribution. Unlike the t-test for the mean, the chi-square test for the variance is not robust to departures from normality, even for large samples. Always verify normality using a probability plot or Anderson-Darling test before applying this test.

Reference: NIST/SEMATECH e-Handbook, Section 1.3.5.8

Formulas

Chi-Square Test Statistic

T=(n−1)s2σ02T = \frac{(n-1)s^2}{\sigma_0^2}

The test statistic scales the sample variance by the hypothesized population variance. Under H0, T follows a chi-square distribution with n-1 degrees of freedom.

Confidence Interval for Variance

(n−1)s2χ1−α/2, n−12≤σ2≤(n−1)s2χα/2, n−12\frac{(n-1)s^2}{\chi^2_{1-\alpha/2,\,n-1}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{\alpha/2,\,n-1}}

The confidence interval for the population variance is constructed by inverting the chi-square test, using upper and lower critical values.