Mean and median

Each value is an ink dot on a number line; equal values stack. The yellow triangle under the line is the mean $\bar{x}$, the dashed line is the median $M$. When you move a value, faint marks record the earlier positions. When an arrow appears, it shows how far the mean moved.

What does the mean balance? Distances. The values to its left lie, in total, exactly as far from it as the values to its right, which is why the triangle works like the fulcrum of a seesaw. Move one value by $\Delta$ and the sum changes by $\Delta$, so the mean moves by $\Delta/n$: drag 7 out to 50 among nine values and the mean shifts by $43/9 \approx 4.78$. However far a value goes, the mean follows.

What does the median balance? Counts. Sorted, it is the middle value, or the midpoint of the middle two when $n$ is even, so at most half the values lie below it and at most half above. As long as the middle value, or the middle two, stay where they are, the median does not move, however far an outer value is dragged. Drag a value through the middle, though, and the median can move with it. To lift it above every original value, the middle value (for even $n$, the upper of the middle two) and every value above it must be changed values above the old maximum. With $n$ fixed, that takes at least $\lceil n/2 \rceil$ changes; with nine values, five.

Why are both called a center? Turn on Try a center and pick a point $c$. The lower plot shows two average distances from $c$ to the values. The solid one, the root mean square distance, satisfies $\mathrm{RMS}(c)^2 = \sigma^2 + (c - \bar{x})^2$: it is lowest at the mean, where it equals the standard deviation $\sigma$ (the variance taken with denominator $n$). The dashed one, the mean absolute distance, is made of straight pieces. Between two values its slope is $(L - R)/n$, where $L$ values lie left of $c$ and $R$ right: stepping right moves you away from the $L$ values and toward the $R$ values. It is lowest wherever neither side has more than half the values, and the median is always one of those places. When $n$ is even and the middle two values differ, that is a flat stretch between them, and the median is its midpoint.