Mean vs Median

A statistic's breakdown point is the fraction of the data that has to be corrupted before the statistic can be made arbitrarily wrong. The mean has a breakdown point of zero: move one single observation far enough and the mean goes wherever you send it. The median has a breakdown point of 50 percent, so you have to corrupt half the dataset before it budges meaningfully. Drag one point off the right edge of the plot above and you can watch both facts at once.

An outlier is one observation sitting far from the rest. Skew is the whole distribution leaning to one side. Both separate the mean from the median, so students often treat them as the same diagnosis, but the remedies differ. An outlier invites you to ask whether that observation is an error. Skew is usually real structure in the data, and reporting a median alongside the mean is the honest response rather than a fix.

• Roughly symmetric with no extreme values: the mean, since it uses every observation and has a smaller standard error.
• Skewed, as with income, house prices, or waiting times: the median, which is why national statistics agencies report median household income.
• Extreme values you cannot justify removing: report both, and say why they differ.
• A compromise when you want most of the data but not the tails: the trimmed mean shown in the readout.