Whenever you collect data, one of the first questions you want to answer is: what is typical? Where does most of the data cluster? Measures of central tendency answer that question by summarising a whole distribution into a single representative value.
There are three main measures, mean, median, and mode, and choosing the right one is not arbitrary. Get it wrong and your summary statistic misrepresents your data. Get it right and you have a solid foundation for everything else in your analysis.
The Mean (Arithmetic Average)
Add up all values and divide by the number of observations. The formula: Mean = Sum of all values ÷ Number of observations.
Example: Five students score 55, 62, 68, 71, and 74 on an essay. Mean = (55 + 62 + 68 + 71 + 74) ÷ 5 = 330 ÷ 5 = 66.
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When to Use It
- Interval or ratio data that is approximately normally distributed.
- When you want a measure that uses every value in the dataset.
- As the basis for further calculations (variance, standard deviation, t-tests).
Its Key Weakness: Sensitivity to Outliers
The mean is pulled by extreme values. Add a sixth student to the example above who scored 12, perhaps they missed weeks through illness, and the mean drops to (330 + 12) ÷ 6 = 57. That is no longer representative of the five students who completed the course normally.
This is why income statistics in the UK use the median. Including one exceptionally high earner in a sample of typical workers inflates the mean but barely moves the median.
The Median
The middle value when all observations are arranged in ascending order. With an odd number of values, it is the value at position (n+1)/2. With an even number, it is the average of the two middle values.
Example: Scores in order: 55, 62, 68, 71, 74. Median = 68 (third value of five). Add the outlier: 12, 55, 62, 68, 71, 74. Median = (62 + 68) ÷ 2 = 65. Barely changed, while the mean fell by 9 points.
When to Use It
- Skewed data – particularly income, house prices, NHS waiting times, and reaction times.
- Ordinal data – you can rank observations but gaps between ranks may not be equal.
- When outliers are present and you want a stable, representative value.
The Mode
The value or category that appears most frequently in a dataset. No arithmetic required, just find what comes up most.
Example: In a group of 20 students, 7 travel by bus, 8 travel by car, 3 walk, and 2 cycle. The mode is ‘car’, the most common mode of transport. You cannot calculate a mean or median for this variable because it is nominal data.
When to Use It
- 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 reporting a single mean.
Comparing The Three Measures
| Measure | Appropriate Data Level | Affected by Outliers | Use When… |
|---|---|---|---|
| Mean | Interval / Ratio | Yes – significantly | Data is normally distributed; you need a basis for further calculations |
| Median | Ordinal, Interval, Ratio | No – resistant | Data is skewed, has outliers, or is ordinal |
| Mode | Nominal, Ordinal, Interval, Ratio | No | Data is categorical; you want the most common value; distribution is bimodal |
In a perfectly symmetrical normal distribution, mean = median = mode. As a distribution becomes skewed, they diverge, and that divergence tells you something important about the shape of your data.
What Happens In Skewed Distributions
| 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 |
The Weighted Mean
Sometimes not all observations contribute equally. A weighted mean assigns a weight to each value before averaging, reflecting relative importance.
Example: A student’s final grade is 40% coursework, 60% exam. Coursework score: 72. Exam score: 55. Weighted mean = (72 × 0.4) + (55 × 0.6) = 28.8 + 33.0 = 61.8. An unweighted average would give (72 + 55) ÷ 2 = 63.5 – 1.7 points higher, which misrepresents how the grade is calculated.
Always Pair Central Tendency With A Spread Measure
A measure of central tendency without a measure of spread is incomplete. Two datasets can have identical means but very different distributions.
- Report mean with standard deviation: M = 66.2, SD = 8.4
- Report median with interquartile range: Mdn = 65.0, IQR = 14.3
- Report mode with frequency and percentage: Modal category = bus (40%, n = 8)
Student Tip: In your dissertation results section, always present descriptive statistics before any inferential tests. A table showing means (or medians) and standard deviations (or IQRs) for each group or variable gives readers the context they need to interpret your statistical tests, and gives examiners confidence that you understand your data before you start testing it.
Frequently Asked Questions
There are three measures of central tendency that help you find the average or the middle value of data. They are mean, mode and median.
Mean: Also called the average, mean is the sum of all values divided by the total number of values. For example, the mean of 2 and 3 will be 2+3 /2 = 2.5
Median: Median is the middle term in a data set. The median in the following data set, 2,3,4,5,6, would be 4 since it’s the value in the center.
Mode: The recurring term in a data set. Mode of the following data set, 1,2,2,3 would be 2 since it appears most times.
The type of measure of central tendency you want to use depends on your data type.
- For nominal data, you can use mode.
- For ordinal data, you can use both mode and median.
- For interval and ratio data, you can use mode, median, and mean.
The most frequently used measure of central tendency is the mean because it gives you the overall average of your data set.
For a single Likert item (e.g., ‘How satisfied are you? 1-5’), use the median and mode. For a composite Likert scale (multiple items summed into a total score), the mean is commonly reported, but acknowledge in your methodology that you are treating the composite score as approximately interval.
It signals skewness. A mean substantially higher than the median indicates positive skew (outliers at the high end pulling the mean up). A mean substantially lower indicates negative skew. Inspect a histogram to confirm, and consider whether the median better represents your data’s centre.
Yes, if every value appears exactly once, there is no mode. This is common in small continuous datasets. In such cases, the mode is not a useful descriptive statistic, and you should rely on the mean or median instead.
For mean and standard deviation: Analyze > Descriptive Statistics > Descriptives. For median, mode, IQR, skewness, and kurtosis: Analyze > Descriptive Statistics > Frequencies (tick the statistics you want in the options). For a comprehensive output including box plots: Analyze > Descriptive Statistics > Explore.
