Karl Pearson's Coefficient of Skewness

Karl Pearson's coefficient of skewness compares the mean and the mode (or, when the mode is unreliable, the mean and the median) relative to the standard deviation.

When the mode is well-defined (a clear, single modal value):

When the mode is ill-defined, absent, or the distribution is bimodal, substitute the empirical relationship Mode $= 3\,\text{Median} - 2\,\text{Mean}$ directly into the formula above. Doing so algebraically simplifies to the equivalent median-based formula:

Both versions are called 'Karl Pearson's coefficient of skewness' — use the mode-based version whenever a genuine, well-defined mode is available (it is the more direct measure), and the median-based version only when the mode cannot be trusted. Because the median-based formula is derived by substituting the empirical mode into the mode-based formula, the two will always agree closely for a moderately skewed unimodal distribution — checking both, where possible, is a good way to catch an arithmetic slip.

Steps to compute Karl Pearson's coefficient for a grouped (continuous) series:

1. Compute the mean $\bar{x}$ using the step-deviation (or direct) method. …