Statistics · Descriptive Statistics
Mean, Median and Mode
The three measures of central tendency — what they are, how to calculate them, and which one to use depending on your data.
When economists analyse income data or teachers review exam scores, they need a single number that summarises a whole dataset. That number is called a measure of central tendency — and there are three of them: the mean, the median, and the mode. Each captures something different about the centre of a dataset, and choosing the wrong one can lead to seriously misleading conclusions.
📘 Key Terms
Mean: The arithmetic average — the sum of all values divided by the number of values.
Median: The middle value when all observations are arranged in order. If there is an even number of values, the median is the average of the two middle values.
Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), two modes (bimodal), or no mode if all values appear equally often.
1. The Mean: Formula and Calculation
x̄ = (Σx) / n
Where Σx is the sum of all values and n is the number of observations.
Example: Exam Scores
Scores: 55, 60, 65, 70, 75, 80, 95
Sum = 55 + 60 + 65 + 70 + 75 + 80 + 95 = 500
Mean = 500 ÷ 7 = 71.4
2. The Median: Formula and Calculation
Step 1: Arrange data in ascending order.
Step 2: Find the middle value.
Step 2: Find the middle value.
For odd n: Median = value at position (n+1)/2
For even n: Median = average of values at positions n/2 and (n/2)+1
Example (same dataset): 55, 60, 65, 70, 75, 80, 95
n = 7 (odd), so median is at position (7+1)/2 = 4th value
Median = 70
3. The Mode: Identification
Example: Survey Responses
Ratings: 3, 4, 4, 5, 4, 5, 3, 4, 5, 4
Mode = 4 (appears 5 times, more than any other value)
When to Use Each Measure
| Measure | Best Used When | Example |
|---|---|---|
| Mean | Data is symmetric, no extreme outliers | Average test score in a class |
| Median | Data is skewed or has outliers | Median household income |
| Mode | Categorical or discrete data; most common value needed | Most popular shoe size |
💡 Key Insight
The mean is highly sensitive to outliers. A single extreme value can pull the mean far from the typical observation. This is why economists report median household income rather than mean household income — a billionaire in a neighbourhood would dramatically inflate the mean while leaving the median unchanged.
⚠️ Common Error
Students sometimes calculate the mean from a frequency table by adding the frequencies and dividing, instead of multiplying each value by its frequency first. Always use: Mean = Σ(f × x) / Σf, where f is frequency and x is the value.
The Effect of Skewness on Measures
In a positively skewed distribution (long tail to the right): Mode < Median < Mean
In a negatively skewed distribution (long tail to the left): Mean < Median < Mode
In a perfectly symmetric distribution: Mean = Median = Mode
Q1. A dataset contains household incomes (£000s): 28, 30, 32, 35, 38, 42, 210. Calculate the mean and median. Explain which better represents the ‘typical’ household income and why. [6 marks]
Answer: Mean = (28+30+32+35+38+42+210)/7 = 415/7 = £59,286. Median = 4th value = £35,000. The median better represents the typical household income because the dataset contains an extreme outlier (£210,000) which pulls the mean upward to £59,286 — far above most households’ earnings. The median of £35,000 is unaffected by this extreme value and better reflects the income of the middle household in the distribution.
References
1. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. 10th edn. W.H. Freeman.
2. Freedman, D., Pisani, R. and Purves, R. (2007) Statistics. 4th edn. W.W. Norton.
3. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. 5th edn. SAGE.
4. Triola, M.F. (2022) Elementary Statistics. 14th edn. Pearson.
5. Blastland, M. and Dilnot, A. (2007) The Tiger That Isn’t: Seeing Through a World of Numbers. Profile Books.