Skewness measures lopsided data, and the textbook rule about it often fails
Many statistics courses teach that in right-skewed data the mean sits to the right of the median. A 2005 journal article pointed out that this rule of thumb fails with surprising frequency, including for something as ordinary as how many adults live in American households.
Skewness describes how asymmetric a distribution is around its mean. In a single-peaked distribution, a long tail stretching to the right gives positive skew and a long tail to the left gives negative skew. The naming trips people up: a left-skewed distribution usually looks like a curve leaning to the right, because the label refers to the direction of the drawn-out tail, not to where the bulk of the data sits.
A tiny example shows the effect. The numbers 49, 50 and 51 are perfectly balanced around 50. Add a low outlier of 40 and the mean drops to 47.5 while the median only slips to 49.5, pulling the data into negative skew. Add a high outlier of 60 instead and the mean jumps to 52.5, above a median of 50.5.
The standard modern measure, Fisher's moment coefficient, averages the cube of each value's distance from the mean in standard units, so large deviations on one side dominate. Any symmetric distribution with a finite third moment, including the normal curve, scores zero, while an exponential distribution scores 2 and a lognormal can take any positive value. The adjusted version used for samples is what spreadsheet and statistics software such as Excel, SPSS and SAS report.
The pitfalls are real. Zero skewness does not prove a distribution is symmetric, because one long thin tail can balance one short heavy tail. And skewness is not tied to the order of mean and median: an older nonparametric definition based on that gap can even disagree in sign with the modern one. The rule breaks most often with discrete data. In US households the distribution of adult residents is skewed to the right, yet most households have no more adults than the median, so the mean falls on the left.
Source: Skewness
