Testing a single variance
Sometimes the spread is the claim: a process engineer guarantees σ = 4, a lab protocol assumes σ² = 16, and the data get a say. The one-sample variance test builds its statistic from the sample variance: χ² = (n − 1)s² ÷ σ0², which follows a chi-square distribution on n − 1 degrees of freedom when the population is normal and the hypothesised variance is right. The interval inverts the same quantiles: (n − 1)s² divided by the upper and lower chi-square cut-offs brackets σ².
A bakery's eight daily totals (42, 38, 45, 51, 44, 39, 47, 50) have s² = 22.571. Against a hypothesised σ0² = 16: χ² = 7 × 22.571 ÷ 16 = 9.875 on 7 degrees of freedom, two-tailed p = 0.3915. No evidence against the claimed variance. The matching 95% interval for σ² runs 9.867 to 93.498, wide, because eight observations pin a variance down only loosely; spread parameters are expensive in data.
The normality warning, taken seriously
Every test on this site leans on assumptions, but this one leans hardest. The chi-square variance test's error rates distort badly under heavy tails: a population with excess kurtosis can double or triple the real false-positive rate while the t-test next door barely notices. If the question is really "do these groups spread differently" or "is this process stable", Levene-type tests and resistant intervals answer it with less fragility. Use the chi-square test when normality is well supported and the single-variance question is exactly the question asked.
Doubling the wrong tail for the two-sided p-value. Chi-square is not symmetric: the two-tailed p is twice the smaller of the two tail areas, the convention this calculator uses, not the area beyond an equal distance on the other side.
Frequently asked questions
Why is the variance test so sensitive to normality?
The chi-square result for (n - 1)s^2 assumes the fourth moment of a normal distribution. Heavy-tailed populations change that moment a lot, and the test's calibration goes with it.
How do I test whether two variances are equal?
That is the F test for two variances (or Levene's test for a sturdier answer). This page handles one sample against a hypothesised value.
What sample size do I need for a variance claim?
More than intuition suggests. In the worked example, n = 8 leaves a 95% interval spanning a factor of nine. Variances are estimated slowly; dozens of observations are typical for tight claims.