Geek hat on.
If exactly half the population are above average, then the dataset is symmetrical about the mean.
If half are above average it means the median is less than the mean.
Here is a bit of 'preaching to the converted' from wiki -
The skewness is not strictly connected with the relationship between the mean and median: a distribution with negative skew can have the mean greater than or less than the median, and likewise for positive skew.
In the older notion of
nonparametric skew, defined as
where
µ is the
mean,
ν is the
median, and
σ is the
standard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than (to the right of) the median, while negative/left nonparametric skew means the mean is less than (to the left of) the median. However, the modern definition of skewness and the traditional nonparametric definition do not in general have the same sign: while they agree for some families of distributions, they differ in general, and conflating them is misleading.
If the distribution is
symmetric, then the mean is equal to the median, and the distribution has zero skewness.
[2] If, in addition, the distribution is
unimodal, then the
mean =
median =
mode. This is the case of a coin toss or the series 1,2,3,4,... Note, however, that the converse is not true in general, i.e. zero skewness does not imply that the mean is equal to the median.
Paul T. von Hippel points out: "Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail in
multimodal distributions, or in distributions where one tail is
long but the other is
heavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal.
[clarification needed] Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median."
[3]
