Measures of Center

A measure of center is simply one number that stands for a "typical" value in the data. The formula $\bar x = \frac{\sum x}{n}$ is the plain old average: $\sum x$ (add up every value) divided by $n$ (how many there are). The only fuss is notation: $\bar x$ when it's a sample, $\mu$ when it's a whole population, same arithmetic. The frequency-table version $\frac{\sum f\,x}{\sum f}$ is the same thing with a shortcut, counting each value $x$ as many times as it appears ($f$).

Median: sort the data; it's the middle value, or the average of the two middle values when $n$ is even. Its position is $\frac{n + 1}{2}$.

Mode: the most frequent value. It's the only center that works for categorical data.

Resistance. Why have two centers at all? Because they react very differently to extreme values. In the incomes $30, 35, 40, 45, 1000$ (thousands), the median is $40$ but the mean is $230$. Think about why. The median only cares about which value is in the middle, so the 1000 could be a million and the median would still be 40. The mean adds up every value, so one huge number gets fully counted and drags the balance point toward it. The median is resistant to outliers; the mean isn't.

Skewness and center: the same tug-of-war explains these.

• Symmetric: mean ≈ median.
• Skewed right: mean > median (the tail pulls the mean up).
• Skewed left: mean < median.

A common mistake. It's tempting to think "average" always means the typical person. With skewed data it doesn't. In the salary example in the practice below, the mean is higher than eight of the nine salaries, so it describes almost nobody.

Governments report median household income and median home price precisely because a few very large values would inflate the mean.