Comparing Distributions
To compare two (or more) data sets, you need a measure of central tendency (what's typical) and a measure of spread (how consistent/variable).
Measures of Central Tendency
Mean
$$\bar{x} = \frac{\sum x}{n} = \frac{\text{sum of all values}}{\text{number of values}}$$
For grouped data, use midpoints:
$$\bar{x} \approx \frac{\sum f \cdot x}{\sum f}$$
Mean is affected by outliers — a very large or small value pulls the mean.
Median
The middle value when data is arranged in order.
- For $n$ values: the median is at position $\dfrac{n+1}{2}$.
- For even $n$: average of the two middle values.
- Median is resistant to outliers.
Mode
The value that appears most often. A data set can have no mode, one mode, or several modes.
Measures of Spread
Range
$$\text{Range} = \text{maximum} - \text{minimum}$$
Simple but affected by outliers.
Interquartile Range (IQR)
$$\text{IQR} = Q_3 - Q_1$$
- $Q_1$ = lower quartile (25th percentile): median of the lower half.
- $Q_3$ = upper quartile (75th percentile): median of the upper half.
IQR measures the spread of the middle 50% — resistant to outliers.
Standard Deviation (Higher)
Measures average distance of each value from the mean. A larger standard deviation means more spread.
Comparing Two Distributions
A full comparison requires:
- Compare a measure of average (state which is higher/lower and what this means in context).
- Compare a measure of spread (state which is more/less consistent and what this means in context).
Exam technique: Always refer back to the context. "Class A has a higher mean score (62 vs 54), showing they performed better on average. However, Class A has a greater IQR (18 vs 10), showing their results were less consistent than Class B."
Choosing the Right Measures
| Situation | Best measure of average | Best measure of spread |
|---|---|---|
| Symmetric data, no outliers | Mean | Standard deviation or range |
| Skewed data or outliers present | Median | IQR |
| Categorical data | Mode | N/A |
WJEC Exam Tips
- On WJEC papers, comparison questions are worth 2–4 marks — one mark per valid comparison, context required.
- Never just write "Class A is better" — always say why (mean/median) and how consistent (range/IQR).
- When asked to "compare", make two separate comparison statements.
- Always state the units in your comparison.
AI-generated · claude-opus-4-7 · v3-wjec-maths