Variance and Chi-Square Tests Online: χ² and F-Tests
What is the One-Variance Chi-Square Test?
The one-variance chi-square test checks whether a population variance equals a target value, using χ² = (n − 1)s²/σ₀² with n − 1 degrees of freedom. It is used to verify that process variation meets a specification or that a measurement system's variance satisfies an allowed tolerance share. A p-value below 0.05 means the variance deviates significantly from the target and the causes of excess variation should be investigated.
When to Use It
Use the F-test to compare the variances of two independent populations, for example two processes or two suppliers, and as the classic variance equality check before ANOVA. Use the chi-square goodness-of-fit test to check whether observed counts follow a theoretical distribution or expected proportions, and the chi-square independence test to analyze whether two categorical variables in a contingency table are related. For data that strongly violate normality, cross-check the F-test with Levene's test.
How to Use It (Step by Step)
Choose the test, then enter either summary statistics (sample size, standard deviation and target variance for the one-variance test) or raw frequency tables for the goodness-of-fit and independence tests. The tool outputs the test statistic, degrees of freedom and p-value. Ensure expected frequencies meet the minimums: at least 80% of expected counts ≥ 5 and all ≥ 1, otherwise merge categories or use Fisher's exact test.
Key Formulas / Example
One-variance χ² = (n − 1)s²/σ₀²; goodness-of-fit χ² = Σ(O − E)²/E; F = s1²/s2² with the larger variance in the numerator. An example independence test is a contingency table of defect type by production line, where a significant χ² indicates the defect mix depends on the line; follow up with standardized residuals to locate the source.