Positive and negative skewed distributions

meansstatistics

I have seen the following statements in several places:

  1. In a normal distribution, mode=mean=median.
  2. In a negative/left-skewed distribution, mode > median > mean.
  3. In a positive/right-skewed distribution, mean > median > mode.

A) I am not clear on why the mode is necessarily affected by skewness. Why is this the case, is this universally true for any unimodal distribution, and/or can you point me to a proof of this?

B) I can sort of see why mean should be pulled in the direction of the skew, but is this universally true for any unimodal distribution and/or is there a proof of this?

Thanks

Best Answer

The mean is generally more sensitive to extreme data or outliers than the mode or median, but the relationships you mention in your points 2 and 3 are not universally true.

However, it is also important to be clear about your definition of skewness. To summarize some essential points from wiki:

-Let $\mu,m,\sigma$ represent the mean, median, and standard deviation respectively for random variable $X$

-There are many measures of skewness; for instance, an old notion of "nonparametric" skewness is defined as $${\mu-m \over \sigma}$$

but the typical modern definition is the third standardized moment:

$$E\left[\left({X-\mu \over \sigma}\right)^3\right]$$

-Under this modern definition:

  1. If the distribution is symmetric, $\mu=m$ and skewness is zero. If the distribution is both symmetric and unimodal, the mode also coincides with $\mu,m$.
  2. For general distributions, there are many examples that break the rules of thumb you mention, as seen here
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