Skewness and Kurtosis

[Giang Lie]

Hi everyone.

I want to compare two samples A and B. The sample size of A is 10 times larger than that of sample B. Both two samples have the same mean but sample A has a larger SD, skewness, and kurtosis (all are positive). Can I make a statement that sample A has a larger part of its observation having a value lower than the mean value compared to sample B? I would really appreciate if anyone can suggest me books or papers that explain how to compare two or more samples based on their statistics.

thanks a lot in advance.

[Nick Cox]

The question on how many values are less then each mean should be answered directly, but suggesting code depends on knowing how you are holding your data. For example, your "samples" A and B could be subsets or they could be held as different variables. Please use dataex to show (an example of) your data.

[Giang Lie]

The question on how many values are less then each mean should be answered directly, but suggesting code depends on knowing how you are holding your data. For example, your "samples" A and B could be subsets or they could be held as different variables. Please use dataex to show (an example of) your data.

Dear Mr. Cox,

thanks a lot for your answer. Do I understand you correctly that my statement about the comparison between the percentage of observations in sample A and B smaller than the mean value is correct? If not, could you please explain more about your answer "The question on how many values are less then each mean should be answered directly". It is hard (at least for me) to provide the fake data representing my case. I want to ask if there is any general rules or applications from which we can extend to special cases when analyzing the data statistics. I am sorry for this very abstract question.

P/S: in my case sample A and B have different observations.

[Maarten Buis]

The basic strategy is:

You can compute the mean of a variable in sample A
you can count how many observations in sample A are below that mean
you can divide that number by the total number of observations in sample A, and you have the proportion less than the mean in Sample A.

repeat for sample B

compare those proportions.

How you implement this basic strategy in Stata, depends on your data.

[Nick Cox]

Maarten Buis gives the advice I would give. Skewness and kurtosis are not a good guide to anything except what they measure directly, being (for example) outluer-prone and also limited (in terms of what estimates can be) by the sample size.

[Giang Lie]

Thank you all for your time and your help. I got the point.

[Giang Lie]

I have another question that I don't know if I get it right. If sample A has a higher value of skewness than that of sample B (both samples are positively skewed), does it mean sample A is more skewed than sample B? And does a very large value of skewness and kurtosis indicate outliers?

[Nick Cox]

If sample A has a higher value of skewness than that of sample B (both samples are positively skewed), does it mean sample A is more skewed than sample B?

Higher skewness means higher skewness — which should sound simple enough, except that are many ways to measure skewness. Skewness in general just means asymmetry of distribution and there are several ways to summarize that, and even an idea that skewness is best treated as a function, not something to be summarized as a scalar.

The moment-based measure is often described as if it were the only measure, but not so. Some are briefly reviewed in Section 7 of https://journals.sagepub.com/doi/pdf...6867X211063415 and there are yet others.

So, it might well be that A is more skew than B on one measure and less skew on another.

And does a very large value of skewness and kurtosis indicate outliers?

Often, but not necessarily. The only way to be clear on this is to define outluers and then check the data against your definition.

[Giang Lie]

Higher skewness means higher skewness — which should sound simple enough, except that are many ways to measure skewness. Skewness in general just means asymmetry of distribution and there are several ways to summarize that, and even an idea that skewness is best treated as a function, not something to be summarized as a scalar.

The moment-based measure is often described as if it were the only measure, but not so. Some are briefly reviewed in Section 7 of https://journals.sagepub.com/doi/pdf...6867X211063415 and there are yet others.

So, it might well be that A is more skew than B on one measure and less skew on another.



Often, but not necessarily. The only way to be clear on this is to define outluers and then check the data against your definition.

Thanks a lot for your detailed answer, Mr. Cox. It helps me a lot.