Mean, Median & Mode

These three numbers are all “averages” — single values that summarize a whole data set — but they answer different questions and behave differently when data is skewed.

Mean

The mean is the familiar arithmetic average: add everything, divide by the count.

Data: 4, 8, 6, 5, 12 → sum = 35, count = 5, mean = 7

The mean uses every value, which makes it sensitive to outliers. Change that 12 to 40 and the mean jumps to 12.6 — even though most of the data did not move.

Median

The median is the middle value when the data is sorted.

Data sorted: 4, 5, 6, 8, 12 → median = 6

With an even count, average the two middle values: for 4, 5, 6, 8 the median is (5 + 6) ÷ 2 = 5.5.

The median resists outliers. In the data 4, 5, 6, 8, 40 the median is still 6, while the mean climbs to 12.6. This is why house prices and incomes are usually reported as medians.

Mode

The mode is the most frequent value.

Data: 2, 3, 3, 5, 7 → mode = 3

A data set can have no mode (all values appear once), one mode, or several. The mode is the only average that works for non-numeric data — the “mode” of red, blue, blue, green is blue.

Choosing the right average

| Situation | Best measure | Why | | | | | | Typical income in a country | Median | A few very high earners drag the mean up | | Average test score | Mean | Every point matters and errors are symmetric | | Most popular shoe size | Mode | You cannot average sizes meaningfully | | Deciding staffing for a hospital | Mean (or distribution) | Total workload matters |

Variance and standard deviation

Once you have a measure of center, you usually want to know the spread. The variance is the average of squared distances from the mean; the standard deviation is its square root, which puts it back in the original units.

Small standard deviation means values cluster near the mean; large means they scatter widely. Together, mean and standard deviation summarize a normal distribution completely.