Mean, Median, and Mode — and When They Disagree

The mean is a balance point: add every value, divide by how many there are. Because every single value gets folded into that sum, one huge or tiny outlier can swing the mean a lot. The median ignores size entirely and just counts position: sort the values and find the middle one. It doesn't care if the biggest value is 40 or 40,000 — it only cares that it's biggest. That's why one billionaire moving into a small town yanks the average (mean) income way up, while the typical (median) income for everyone else barely shifts. The mode just names whatever value shows up most often, which has nothing to do with size or position at all.

This is why skew matters. In a data set with a long tail stretching toward large values, the mean gets pulled toward that tail, landing higher than the median. The median sits still because it only counts its way to the middle position, unaffected by how extreme the tail gets. On a symmetric data set, mean and median land close together, so either one is a fair summary; on a skewed set, the median is the honest description of "typical," and the mean is inflated or deflated by the outliers.

This topic connects to shape: a mean pulled above the median signals right skew, and a mean below the median signals left skew, which is exactly how the test expects you to read a distribution's shape without seeing a picture of it. It also sets up weighted averages, where each value doesn't count equally — a natural next step once you're comfortable with the plain mean.