# How to test if a distribution is symmetric?

How to test if a distribution is symmetric??

Hi all:
To test if a distribution is symmetric or not, I knew we can use the
mean-median == 0
and
skewness == 0
I am wondering if there is any other methods of doing so? Also, which one of them are more sensitive to the data changes please? I mean if I slightly change some data in order to destroy the symmetric, which way is more sensitive to detect the changes please?
Thanks

## Answers and Replies

JasonRox
Homework Helper
Gold Member
Hi all:
To test if a distribution is symmetric or not, I knew we can use the
mean-median == 0
and
skewness == 0
I am wondering if there is any other methods of doing so? Also, which one of them are more sensitive to the data changes please? I mean if I slightly change some data in order to destroy the symmetric, which way is more sensitive to detect the changes please?
Thanks

I would say if the Variance is low then small data change can through everything off. If the Variance is high, data change doesn't really do much since it's already all over the place.

That's my guess. I know nothing about this stuff.

Also, to check if it is symmetric, I would assume if f(x) is your distribution function then f(x)=f(-x) tells us it is symmetric.

D H
Staff Emeritus
Science Advisor
Pearson's skewness coefficients involve mean-mode and median-mode. You mention skewness itself. There are plenty of other measures out there.

A truly symmetric distribution will have zero values for all odd moments about the mean. Just because a certain distribution has zero skewness does not necessarily mean it is symmetric. The problem with moments higher than order 3 or 4 is that the values obtained for such moments from any realistically gathered dataset are highly suspect. Bottom line: stick with lower moments (the standard skewness coefficient or Pearson's skewness coefficient).

For any skewness coefficient, you cannot simply test whether the result you obtain is zero or not. You need to test whether the result you obtain differs from zero in a statisically meaningful way.