Measures of Central Tendency
Mean, median and mode all answer the same question — "where's the centre?" — but they answer it in different ways, and knowing which to trust is as important as knowing how to calculate them.
Mode, median and mean
A measure of location is a single value that describes a position within a dataset. When that value describes the centre of the data, it's called a measure of central tendency. There are three in regular use:
Mode suits qualitative data, or quantitative data with a clear single (or double, "bimodal") peak — it's uninformative if every value occurs only once. Median suits quantitative data, especially when extreme values are present, since it isn't affected by them. Mean also suits quantitative data and uses every value, giving a genuinely representative average — but that same property means it's pulled around by extreme values.
The mean from a frequency table
When discrete data is presented in a frequency table, the mean is found using
where $\sum xf$ is the sum of each data value multiplied by its frequency, and $\sum f$ is the total frequency (equivalently, $n$).
Estimating the mean from grouped data
When continuous data has been grouped into classes, the individual values are no longer known — so every value in a class is treated as if it sat exactly at that class's midpoint, giving an estimate of the mean:
where $x$ now stands for each class midpoint. This is always described as an estimate, since the true values within each class are unknown.
Grouping data means you know how many observations fall inside a class, but not exactly where. Using the midpoint assumes an even spread across the class — a sensible approximation, but not a fact about the data. Any statistic calculated from grouped data inherits this same uncertainty.
Worked example 1
Always order the data before finding the median — applying the position formula $\frac{n+1}{2}$ to unordered data is one of the most common, and entirely avoidable, errors.
Worked example 2
| Number of pets | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency | 9 | 16 | 11 | 4 |
Find the mean number of pets per household.
Worked example 3
| Delivery time (min) | 10–19 | 20–29 | 30–39 | 40–49 |
|---|---|---|---|---|
| Frequency | 6 | 21 | 17 | 6 |
Every parcel in the "10–19" class is treated as if it took exactly $14.5$ minutes — none actually did. That approximation is unavoidable once data is grouped, which is why the answer must be reported as an estimate.
Worked example 4
Whenever a dataset is skewed by one or two extreme values — house prices, incomes, company salaries — the median gives a far more representative "typical value" than the mean, which is dragged toward the extreme end.
For grouped data, the mode is usually reported as the modal class — the class with the highest frequency — since no individual value inside a class is known exactly. Don't report a single number as "the mode" of grouped data unless the question specifically asks for the midpoint of the modal class.
Check yourself
Find the mean and median of the dataset: $14, 18, 11, 25, 18, 21, 13$.
(Hint: order the data first before finding the median's position.)