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F-Test - Two Variables


The F-Test is used to compare the variances of two populations on a continuous CT characteristic. Since we don’t know the population variances, an analysis of two samples of data is required.

This test is usually used to determine if there is a statistically significant change in the variance of a CT characteristic under two conditions.



Reference: Juran’s Quality Control Handbook


A. Null (H0) and alternative (Ha) hypotheses, where the two population variances are compared. For the alternative hypothesis, one of the three hypotheses has to be chosen before collecting the data to avoid being biased by the observations of the samples.

B. Excel output for an F-Test with directional alternative hypothesis

C. Descriptive Statistics - sample mean, sample variance, number of observations or sample size.

D. Number of degrees of freedom (df = number of observations – 1, for each sample).

E. Computed or observed F statistic ( , see tool Distribution – F).

F. P-Value – This value has to be compared with the alpha level and the following decision rule is used: if P < alpha, reject H0 and accept Ha with (1-P) 100% confidence; if P > alpha, don’t reject H0.

G. Tabulated Fisher (F) distribution value with alpha risk

Major Considerations

The assumptions for using this test is that the data comes from two independent random samples taken from two normally distributed populations.

This test is sensitive to the normality assumption. It is available in Excel but not in Minitab where the function Homogeneity of Variance under Stat>ANOVA can be used instead (see tool Homogeneity of Variance Tests).

Application Cookbook

1. Define problem and state the objective of the study.

2. Establish hypothesis - State the null hypothesis (H0) and the alternative hypothesis (Ha).

3. Establish alpha level, usually is 0.05.

4. Select random samples.

5. Measure the CT characteristic.

6. Analyze data with Excel:

– When the alternative hypothesis is directional (> or <), use the function under Tools>Data Analysis> F-Test Two-Sample for Variances (If you do not have a Data Analysis option, select the following under Tools>Add-Ins>Analysis ToolPak).

Put data in two columns. After selecting the function, input variable 1 & 2 ranges. Input alpha level (default=0.05) and select output range.

Conducting and F-Test

– When the alternative hypothesis is non directional, use the statistical function (fx) called FTEST. The only output from this function is the P-value.

7. Make statistical decision based on the output from Excel. Either accept or reject H0. Translate statistical conclusion to practical decision about the CT characteristic.

From F-Test to Six Sigma Tools.

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