"Average" can mean mean, median, or mode—choose carefully
An average is a value meant to sit at the center of a collection. People usually mean the arithmetic mean—sum divided by count—but teachers also lump in the median and mode. For skewed incomes, the median often tells the truer everyday story.
In mathematics an average is a central, common or typical summary of a group. Most often that is the arithmetic mean: add the numbers and divide by how many there are, so [2, 3, 4, 7, 9] averages to 5. Sample means (from observed subsets) are distinguished from the underlying distribution's expected value. A mean always lies between the extremes; if every entry is identical, every average equals that value. Common averages are monotonic under coordinated increases, linearly homogeneous under scaling, and unchanged by reordering.
Other "averages" answer different questions. The median—the middle value after sorting—resists outliers, which is why personal income is usually reported as median income rather than a mean inflated by a few fortunes. The mode is the most frequent value, useful for categorical data or histograms. The harmonic mean, the reciprocal of the mean of reciprocals, fits rates over equal distances: it equals the constant speed that covers the same total distance in the same time. Geometric and quadratic means, the mid-range, and generalized f-means (with invertible f) appear less often under the everyday word "average."
Statisticians package these ideas as measures of central tendency—a phrase dating from the late 1920s—contrasted with dispersion. For skewed data the mean, median and mode can diverge sharply: mean income sits high, median splits the population in half, mode favours the crowded lower pile. Yet some skewed laws, such as exponential and Poisson, are still best summarized by their means. For unimodal distributions, sharp inequalities bound how far mean, median and mode can separate relative to the standard deviation—for example the mean–median gap over σ is at most 1 in one classic bound, with tighter or related √3 and √0.6 forms involving the mode.
Source: Average
