CFA
For the positively skewed unimodal distribution, the mode is less than the median, which is less than the mean. For the negatively skewed unimodal distribution, the mean is less than the median, which is less than the mode.
(Institute 423)
Institute, CFA. 2016 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. CFA Institute, 07/2015. VitalBook file.
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Wikipedia
The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, and likewise for positive skew.
Place your bets ladies and gentlemen.
Wikipedia doesn’t restrict it to unimodal distributions.
CFA is still wrong. It can fail to hold in unimodal distributions.
(See figure 2 if you want just one example…)
https://ww2.amstat.org/publications/jse/v13n2/vonhippel.html
It looks that discrete distributions were really discrete with their secret…
Debunked.
If I recall, it can happen in continuous distributions as well, but it’s more common in the discrete distributions.
Thanks for the links guys, enlightening stuff :).
Just a question though, does the kurtosis formula still work with these exceptions? Can you compare the kurtosis result from two diagrams one holding true to the rule of thumb and one that suggests the rule of thumb doesn’t work?
Another interesting note:
In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution, but is also true for an asymmetric distribution where the asymmetries even out, such as one tail being long but thin, and the other being short but fat.
That’s taken from wikipedia again.