The three measures of central tendency are the mean, the median and the mode. Each summarises a whole dataset into a single representative value that tells you what is “typical”: the mean is the arithmetic average, the median is the middle value, and the mode is the most frequent value. This guide works through all three step by step on the same dataset, gives you the formulas, and shows exactly when to use each one.
To make the comparison concrete, every worked example below uses one shared dataset:
Choosing the right measure is not arbitrary. Pick the wrong one and your summary statistic misrepresents the data; pick the right one and you have a solid foundation for the rest of your analysis. Whenever you collect data, central tendency is one of the first things you should report.
1. The Mean (Arithmetic Average)
The mean is what most people call the average. You add up all the values and divide by the number of observations.
Formula: Mean (x̄) = (Sum of all values) ÷ (Number of values) = Σx ÷ n
- Step 1 — add the values: 4 + 8 + 6 + 5 + 3 + 8 + 7 = 41
- Step 2 — count the values: n = 7
- Step 3 — divide: 41 ÷ 7 = 5.86 (2 d.p.)
The mean score is 5.86.
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When to Use the Mean
- Interval or ratio data that is roughly symmetrical (normally distributed).
- When you want a measure that uses every value in the dataset.
- As the basis for further calculations such as variance, standard deviation and t-tests.
Its Key Weakness: Sensitivity to Outliers
The mean is pulled towards extreme values. If one student in our dataset had scored 30 instead of 7, the sum would jump to 64 and the mean to 64 ÷ 7 = 9.14 — higher than six of the seven actual scores. That is why income statistics in the UK are usually reported using the median: a single very high earner inflates the mean but barely moves the median.
2. The Median (The Middle Value)
The median is the middle value once all observations are arranged in order. It splits the dataset in half: 50% of values fall below it and 50% above.
Formula: with an odd number of values, the median is the value at position (n + 1) ÷ 2. With an even number, it is the mean of the two middle values.
- Step 1 — sort ascending: 3, 4, 5, 6, 7, 8, 8
- Step 2 — find the middle position: (n + 1) ÷ 2 = (7 + 1) ÷ 2 = 4th value
- Step 3 — read off the 4th value: 6
The median score is 6. (If there were an even number of values, you would average the two middle ones — for example, the median of 3, 4, 5, 6 is (4 + 5) ÷ 2 = 4.5.)
When to Use the Median
- Skewed data — particularly income, house prices, NHS waiting times and reaction times.
- Ordinal data — you can rank observations, but the gaps between ranks may not be equal.
- When outliers are present and you want a stable, representative value.
3. The Mode (The Most Frequent Value)
The mode is the value or category that appears most often in a dataset. There is no arithmetic involved — you simply find what occurs most frequently. A dataset can have one mode, two modes (bimodal), several, or none at all if every value occurs only once.
- Step 1 — tally how often each value appears: 3×1, 4×1, 5×1, 6×1, 7×1, 8×2
- Step 2 — pick the most frequent: 8 occurs twice; every other value occurs once
The mode is 8. The mode also works for non-numeric data: in a survey where 8 students travel by car, 7 by bus, 3 walk and 2 cycle, the mode is “car” — and for nominal data like this, the mode is the only valid measure of central tendency.
When to Use the Mode
- Nominal data — the only appropriate measure of central tendency.
- Any time you want to know the most common response or category.
- Bimodal distributions — when data has two peaks, reporting both modes is more informative than a single mean.
4. Mean vs Median vs Mode: Comparison Table
On our shared dataset (3, 4, 5, 6, 7, 8, 8) the three measures give mean = 5.86, median = 6, mode = 8. The table below summarises how they differ and when to choose each.
| Measure | Definition | Best Used When | Affected by Outliers? | Data Level |
|---|---|---|---|---|
| Mean | Sum of all values divided by the number of values | Data is roughly symmetrical and you need a basis for further calculations | Yes — strongly | Interval / ratio |
| Median | The middle value when data is ordered | Data is skewed, has outliers, or is ordinal | No — resistant | Ordinal / interval / ratio |
| Mode | The most frequently occurring value or category | Data is categorical, or you want the most common value | No | Any (incl. nominal) |
In a perfectly symmetrical normal distribution, mean = median = mode. As a distribution becomes skewed they diverge, and that divergence itself tells you about the shape of your data.
| Distribution | Relationship | Practical Example |
|---|---|---|
| Normal (symmetric) | Mean = Median = Mode | Standardised test scores designed to be normally distributed |
| Positive skew (right tail) | Mode < Median < Mean | Income data, house prices, NHS waiting times |
| Negative skew (left tail) | Mean < Median < Mode | Easy test scores; age at death in high-income countries |
5. The Weighted Mean
Sometimes not all observations contribute equally. A weighted mean assigns a weight to each value before averaging, reflecting its relative importance.
- Apply the weights: (72 × 0.4) + (55 × 0.6) = 28.8 + 33.0
- Weighted mean: 61.8
An unweighted average would give (72 + 55) ÷ 2 = 63.5 — 1.7 points higher, misrepresenting how the grade is actually calculated.
6. Always Pair Central Tendency With a Spread Measure
A measure of central tendency without a measure of spread is incomplete. Two datasets can share an identical mean yet have very different distributions, so always report central tendency alongside a measure of variability — the range, interquartile range or standard deviation.
- Report the mean with the standard deviation: e.g. M = 66.2, SD = 8.4
- Report the median with the interquartile range: e.g. Mdn = 65.0, IQR = 14.3
- Report the mode with frequency and percentage: e.g. modal category = bus (40%, n = 8)
Together, central tendency and spread form the backbone of descriptive statistics.
