Mean vs Median: One Big Salary

Ten people at a small company earn between $28k and$ 79k. Add the founder's salary and the mean jumps while the median barely moves. The median and IQR resist outliers; the mean and standard deviation chase them.

How it works

Measures of center and spread, and resistance to outliers

In one sentence

The mean $\bar x = \frac{1}{n}\sum x_k$ and the standard deviation $s$ depend on the size of every value; the median $\tilde x$ and the interquartile range $Q_3 - Q_1$ depend only on the order of the values, which makes them resistant to outliers.

General formula

{\htmlData{sym=0}{\bar x}} = \frac{1}{\htmlData{sym=3}{{n}}}\sum {\htmlData{sym=2}{x_k}}, \quad {\htmlData{sym=1}{\tilde x}} = \text{middle value}, \quad {\htmlData{sym=4}{s}} = \sqrt{\frac{\sum ({\htmlData{sym=2}{x_k}} - {\htmlData{sym=0}{\bar x}})^2}{{\htmlData{sym=3}{n}}-1}}, \quad {\htmlData{sym=7}{\text{IQR}}} = {\htmlData{sym=6}{Q_3}} - {\htmlData{sym=5}{Q_1}}

The mean is the balance point, so a far-away weight tips it. The median just counts how many values sit on each side.

What the symbols mean

$\bar x$
Mean
The sum of the salaries divided by how many there are.
$\tilde x$
Median
The middle salary when they're sorted (the average of the two middle ones for an even count).
$x_k$
One salary
The kth person's salary, in thousands of dollars.
$n$
Number of salaries
Ten, or eleven with the founder.
$s$
Standard deviation
The typical distance of a salary from the mean.
$Q_1$
First quartile
A quarter of the salaries are at or below it.
$Q_3$
Third quartile
Three quarters of the salaries are at or below it.
$\text{IQR}$
Interquartile range
The width of the middle half of the salaries.

The mean $\bar x$ adds every salary and divides by $n$ , so one huge salary pulls it hard.
The median $\tilde x$ is the middle salary once they're sorted: one extreme value moves it by at most half a step.
The standard deviation $s$ measures the typical distance from the mean, with distances squared, so an outlier inflates it. The IQR, $Q_3 - Q_1$ , spans the middle half and ignores the extremes.
For skewed data like incomes, report the median and IQR; for roughly symmetric data the mean and standard deviation work well.

Learn the math

Measures of CenterStats · Lesson 6, Descriptive Statistics Measures of SpreadStats · Lesson 7, Descriptive Statistics