skewness
Some distributions are lopsided. Incomes, house prices, and waiting times have a wall on the left (you cannot earn negative money) and a long thin tail stretching to the right, where a few enormous values live. Skewness is the single number that measures this lopsidedness — which way a distribution leans and how strongly.
It is built from the third central moment, standardized by the cube of the standard deviation so it has no units: skewness = E[(X - mu)^3] / sigma^3. The cube preserves sign, so the sign tells the direction. Positive (right) skew means a long right tail — the bulk of the data sits on the left and a few big values pull the right tail out; for such a distribution the mean is dragged above the median. Negative (left) skew is the mirror image, with a long left tail and the mean below the median. A symmetric distribution like the normal has skewness exactly 0.
Skewness flags when 'the average' may be a misleading summary. With strong right skew, the mean can sit well to the right of the typical value, so the median is often the more honest centre — this is why median household income is reported, not mean. Two cautions: skewness is very sensitive to outliers because of the cube, and a skewness of 0 does not guarantee symmetry — it only means the third central moment cancels out, which a non-symmetric distribution can also achieve.
Household income is strongly right-skewed: most households cluster at modest incomes, but a few very high earners stretch the right tail. The result is mean income well above median income — so the 'average' overstates what a typical household earns.
Positive skew: long right tail, mean pulled above the median.
Skewness 0 does not prove symmetry — it only means the third central moment vanishes, which some non-symmetric distributions also satisfy. And because of the cube, a single outlier can swing the value dramatically.