# What is the difference between “scale parameter” and the variance?

I would like to understand the difference between "scale parameter" and the variance of a distribution? I found, that the "scale parameter", scaling the width of a distribution is mentioned when speaking about nonparameteric methods, while the variance and moments are called when speaking about parametric methods. Sometimes this is also referred as dispersion. When the dispersion grows, the scale grows, so does the variance. I understand that dispersion is some general concept, so probably the scale parameter, and the variance is a special measure of dispersion and scale?

When I read a documentation of statistical methods for comparing variances, I can see that scale is equated somehow with the variance. But when I run both tests for equality of variances and scales on the same data, I often get contradicting results, as in the linked example. Because I am asking in the context of this question: Why do we need F test of two variances if we have the Ansari-Bradley test? Knowing the answer I could, probably, answer the linked question.

• You are correct: scale parameter is indeed the more general term. Often the scale parameter doesn't even have the same dimensions as variance, let alone be equal to it. A fairly simple example is a two-parameter gamma distribution. One parameterisation has a scale parameter with the same units and dimensions as the original variable, but the SD and variance depend on both scale and shape parameters. – Nick Cox May 3 at 14:25
• Thank you very much for your response. I apologize for drilling this topic, but would you be willing, kindly please, to tell me more about the linked question? It relates to this discussion and is about the tests of scales and variances. There is an example, where the test of scales reports non-significance, while the test of variances contradicts it. In the light of what you said, it seems, that variance, as a "special case" should theoretically agree (if one grows, the other should too), but it may happen, that scales are similar, but variances differ?Is there any "formula" for scale? – Katikarnata May 3 at 14:37
• Sorry, no. To answer that it would be necessary (and might not be sufficient) for me to study what the Ansari-Bradley test does and at the moment I lack the time and the inclination. But it's no surprise to me that different tests looking for different things can appear to contradict each other. – Nick Cox May 3 at 15:13

“Variance“ has a definite meaning. Variance always means the second central moment, and when we estimate or test the variance, we are estimating or testing this quantity.

“Scale” is more general. It refers to spread of the data in some way but without committing to discussing the second central moment. After all, the second central moment might not exist!

I like the definition I’m seeing on Wikipedia for a scale parameter $$s$$ (and other parameters $$\theta$$):

$$F(x; s, \theta)=F(x/s;1,\theta)$$

So if some $$s$$ allows us to stretch or compress the CDF to some standardized CDF, we call it a scale parameter. It might be related to the variance, but maybe not.

https://en.wikipedia.org/wiki/Scale_parameter

• Thank you very much. Yes, I saw this definition, which told me why it's called "scale" (it scales the "width" of the distribution). When the dispersion grows, the scale does too, so the variance should too. So, I thought the two properties of a distribution should agree. But in the example that I linked, the Ansari-Bradley test of scales shows no difference, while the test of variances shows difference at very low p-value. From this I concluded those measures aren't the same, but in documentations it is mentioned, that the A-B test verifies equality of variances. That's why I'm so confused! – Katikarnata May 3 at 14:41
• @Katikarnata I think I see what you mean in the Ansari-Bradley documentation. It’s the “notes”, right? Notice the mention of normality. Then the scale parameter is standard deviation which has a known relationship to variance. – Dave May 3 at 14:51
• That's it! Thank you. So, now I understand this test is "more general" (based on ranks), which - in case of the normal distribution, "converges" (I know it's bad word, but in layman's terms) to comparing variances. I run a simulation and sampled from the normal distribution. Now the outcomes agree, but sometimes it's on the border line, for example: [AB = 5490, p-value = 0.03153 vs. F = 0.36666, p-value = 0.000001059] and most of the time AB "judged" more liberally. But I believe it's the nature of a non-parametric test, which is usually "weaker", not making distributional assumptions, right? – Katikarnata May 3 at 15:13
• @Katikarnata When I took nonparametric methods, the professor’s comment was that a good parametric test should be almost as good as the parametric test when the parametric test’s assumptions are met and beat the heck out it when those assumptions are violated. But that comment is a concession that the nonparametric method would be worse if the assumptions are met, even if only slightly worse. – Dave May 3 at 15:18
• @Dave I think a non- is missing in your comment. – Nick Cox May 3 at 17:58