## Tuesday, August 11, 2015

### Symmetry and Skewness

After taking your first introductory course in statistics you probably agreed wholeheartedly with the following statement:
"A statistical distribution is symmetric if and only if it is not skewed."
After all, isn't that how we define "skewness"?

In fact, that statement is incorrect. There are distributions which have a skewness coefficient of zero, but are asymmetric.

Before considering some examples of this phenomenon, let's take a closer look at the meaning of "skewness" in the statistical context.

Suppose that a random variable, X, has a mean of μ and a standard deviation of σ. Then the third standardized moment of X is defined as γ= E[(X - μ) / σ]3, and this is a very common measure of skewness. Clearly, the skewness coefficient, γ1, can be positive, negative, or zero.

Also, note that γ1 may not be defined at all - this will be the case if any of the first three moments of the distribution of X are undefined. An obvious example of this is the Cauchy distribution, none of whose integer-order moments are defined. (The associated integrals diverge.)

While γ1 is "a" common measure of skewness, it's not the only one that people use.

Students are often taught to use (Karl) Pearson's measure of skewness - namely Skewness = (mean - median) / (std. deviation). In this case we have positive skewness if the mean exceeds the median, and negative skewness if the converse is true. However, this relationship between the mean and median doesn't necessarily determine the sign of γ1, especially in the case of discrete distributions. von Hippel (2010) provides an example of this, and he also discusses several other skewness measures - especially ones that are more robust to outliers than is γ1.

We should also make sure that we understand what we mean by "symmetric" when we're talking about statistical distributions. Meijer (2000) states the conventional definition as follows:
"A random variable X is symmetrically distributed if there exists a constant µ such that X − µ and −(X − µ) have the same distribution and is asymmetrically distributed if there does not exist such a constant. The distribution and density (or probability mass function) of X are also accordingly called symmetric or asymmetric".
To illustrate symmetry, think about the case where X is Normal (μ , σ2).Then both (X − µ) and −(X − µ) are Normal(0, σ2), so X has a symmetric distribution. (Of course, you knew that already,)

Now, what about the connection between skewness and the symmetry of the distribution?

First, when γ1 is finite, its value will be zero if the distribution is symmetric.

Second, if γ1 is finite but non-zero, then the distribution of X must be asymmetric.

However........ a distribution can be asymmetric but have γ1 = 0.

In other words, zero skewness is necessary, but not sufficient, for the symmetry of a probability distribution.

Ord (1968) provides an example of this last result, in the form of the discrete Student's t distribution. Meijer (2000) shows that a mixture of K normal random variables can also have a distribution which is asymmetric, even though γ1 = 0.

For instance when K = 2, µ1 = −2, µ2 = 1, σ12 = 1, σ22 = 2, and the mixing weights are 1/3 and 2/3, then E(X) = E(X3) = 0. This distribution has zero skewness but it is asymmetric. Its density function appears below:

References

Meijer, E., 2000. An asymmetric distribution with zero skewness. Mimeo., University of Groningen.

Ord, J. K., 1968. The discrete Student's t distribution. Annals of Mathematical Statistics, 39, 1513-1516.

von Hippel, P., 2010. Skewness. In M. Lovric, ed., International Encyclopedia of Statistical Science. Springer, New York.