Mean, median and mode: how to calculate them and when to use each

Mean, median and mode are measures of central tendency: each summarizes a data set with a single typical value. They answer different questions, and choosing the wrong one can distort the conclusion.

Arithmetic mean

Worked example: grades 6, 7, 7, 8 and 9

1. Sum: $6 + 7 + 7 + 8 + 9 = 37$.
2. Count: $n = 5$.
3. Mean: $\dfrac{37}{5} = 7.4$.

Weighted mean

When each value has a weight $w_i$:

Worked example: grade 6 with weight 2 and grade 8 with weight 3

1. Weighted sum: $2 \cdot 6 + 3 \cdot 8 = 12 + 24 = 36$.
2. Sum of weights: $2 + 3 = 5$.
3. Mean: $\dfrac{36}{5} = 7.2$.

Median

Sort the data. With an odd $n$, the median is the middle value. With an even $n$, it is the mean of the two middle values.

Worked example: 9, 4, 7, 10, 8

1. Sort: $4, 7, 8, 9, 10$.
2. $n = 5$, odd: the 3rd value is in the middle.
3. Median: $8$.

Worked example: 10, 4, 8, 7

1. Sort: $4, 7, 8, 10$.
2. $n = 4$, even: the middle values are $7$ and $8$.
3. Median: $\dfrac{7 + 8}{2} = 7.5$.

Forgetting to sort is the most common median mistake.

Mode

It is the value that repeats most. In $3, 5, 5, 6, 9$, the mode is $5$. In $2, 2, 4, 4, 7$, there are two modes ($2$ and $4$): the set is bimodal.

The mode is the only one of the three that works with categorical data: if “bus” is the most cited means of transport in a survey, it is the mode. There is no mean of means of transport.

When to use each one

Worked example: salaries at a small company (in thousands): 2, 2, 3, 3, 3, 4, 20

1. Mean: $\dfrac{37}{7} \approx 5.29$.
2. Median: there are 7 values, and the 4th is $3$.
3. Mode: $3$.
4. Six of the seven people earn 4 thousand or less. The mean of 5.29 thousand is pulled up by the 20 thousand salary and does not represent the typical employee; the median and the mode do.

Situation	Best measure
Symmetric data without outliers	mean
Outliers or skewed distribution (income, home prices)	median
Categorical data or the most common value (best-selling clothing size)	mode
Values with different importance (weighted grades)	weighted mean

Common mistakes

1. Computing the median without sorting the data.
2. Using the mean to describe income or prices with a few very high values.
3. In a weighted mean, dividing by the number of values instead of the sum of the weights.