Skewness and kurtosis, and where the definitions split
Skewness measures how lopsided a distribution is and kurtosis measures how peaked it is and how heavy its tails are. Positive skewness means a longer right tail and negative skewness a longer left tail. The catch is that neither quantity has a single formula, so this calculator makes you choose the basis and then prints the chosen definition in the result labels and in a formula row.
The first fork is sample versus population. The population basis uses the raw moments, dividing the third moment by the cube of the standard deviation. The sample basis applies the n/((n−1)(n−2)) correction, producing the adjusted Fisher-Pearson coefficient that matches Excel's SKEW and KURT. The two values differ on identical data, so any report should state which one it uses.
The second fork is whether 3 is subtracted from the kurtosis. A normal distribution has a kurtosis of 3, so software usually reports excess kurtosis, which is that value minus 3. Positive excess kurtosis means a sharper peak than normal, described as leptokurtic, while negative excess kurtosis is flatter, or platykurtic. Skewness needs at least 3 values and kurtosis at least 4, and because sample kurtosis divides by (n−2)(n−3) it cannot be computed from only 3 values. If every value is identical the standard deviation is 0 and neither quantity is defined.
Frequently Asked Questions
Population skewness divides the third moment by the cube of the standard deviation. Sample skewness adds the n/((n−1)(n−2)) correction, giving the adjusted Fisher-Pearson coefficient that Excel returns from SKEW.
A normal distribution has a kurtosis of 3, so subtracting 3 gives excess kurtosis. Above zero the distribution is more peaked with heavier tails than normal, and below zero it is flatter.
Skewness needs at least 3 values and kurtosis needs at least 4. The sample kurtosis formula divides by (n−2)(n−3), so with only 3 values it cannot be computed.