median vs. mean tendencies Would there be an instance  of a variable for which the mean would be a more meaningful or appropriate descriptor of central tendency than the median? And vice-versa of an example of a variable for which the median would be a more meaningful or appropriate descriptor than the mean? 
I thought about central distributions and the weight of the cuve
 A: I believe there are various points of view on this.
One is to say that the median is a more appropriate measure of location for a distribution (hence also sample) that is strongly skewed.
For data, it is a good idea to consider what is actually important for the purpose at hand. Consider
salaries at a company where there are many workers being paid near minimum wage and a few in various levels of management being paid orders of magnitude more.
A sociologist wanting to focus on what affects the most people or the 'typical'
person might use the mode or median do describe the location of this right-skewed
distribution. By contrast, the CFO might argue in favor of the mean because
(s)he is focused on total payroll, which is the number of employees times the mean.
Often (not invariably) for a strongly right-skewed distribution, the mode
is smallest, the median is larger, and the mean (center of gravity pulled
rightward by the long tail) is largest.
Members of the gamma family of distributions, parameterized by a shape parameter $\alpha$ and a rate parameter $\beta$ are generally
right-skewed. (See Wikipedia on 'gamma distribution') When $\alpha > 1,$
there is a mode at $(\alpha - 1)/\beta,$ the mean is $\alpha/\beta,$
and the median (found by numerical integration) is between. For example,
here is a graph of the PDF of $Gamma(\alpha=3, \beta = 2).$ Vertical red, purple,
and blue lines show positions of the mode (at 1), median (at 1.33703), and mean (at 1.5), respectively. 
qgamma(.5, 3, 2)  # median
## 1.33703


From a more theoretical point of view, if you prefer maximum likelihood
estimators (MLEs), then there are times when the center of a symmetrical
distribution is best esstimated by the sample mean and other times when the sample median is best.
For a normal distribution the sample mean $\bar X$ is the MLE of the
center $\mu$ (mean and median) of the distribution. By contrast for a
Laplace (double exponential) distribution the sample median $\check X$ is the MLE of the
center $\eta$ (mean and median). Here an estimator is 'best' if it has
the smallest variance, thus giving the most reliable estimate.
