# How is it possible that these variances are equal?

I'm using the Fligner-Killen test to analyze the residuals of a linear regression. I subdivide those residuals in three groups and then I do the FK test to check the homogeneity of variances.

The result is:

fligner.test(pair.res, pair.groups)

Fligner-Killeen test of homogeneity of variances

data:  pair.res and pair.groups
Fligner-Killeen:med chi-squared = 2.6937, df = 2, p-value = 0.2601


In the image below I have plotted the residuals and the groups. Could someone explain me WHY those three groups have the same variance? It does not seem correct doing a simple visual check.

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A good visual test to estimate or compare standard deviations (after checking for obvious outliers) is to look at the range of a dataset. For a given sample size, the range will typically be near a fixed multiple of the SD. With around 250 independent samples of a normal distribution, for instance, the range will be around 7 times the SD. So here we have ranges of around 1.3 (left panel), 1.0 (middle panel), and 1.1 (right panel), and each panel comprises about the same amount of data. Thus the ratios of variances, which will equal the squares of the ratios of the ranges, range from around $(1.3 : 1.1)^2$ = about $1.4$ down to $(1.0 : 1.1)^2$ = about $0.8$. You could use an F-test as a very rough estimate of significance, but I would reduce the degrees of freedom to account for the evidently strong serial correlation. Regardless, it wouldn't be unusual to get a pair of F-statistics (with 250 and 250 degrees of freedom) in the range $(0.8, 1.4)$ and reducing the df only increases that chance. In light of that, a p-value of $0.26$ looks fine.