Comparing statistical distributions means deciding which of two or more data sets is "better" or "different" by examining their average (a measure of location) and their spread (a measure of variation). At GCSE, a full comparison always requires one comment about the average and one about the spread, supported by specific numerical evidence from the data.
Why do you need two measures to compare distributions?
Comparing only the average can be misleading. Two groups might have the same mean but very different spreads — one group may be consistently near the average while the other has extremely high and low values. A full comparison needs:
- A measure of location (average): mean, median, or mode — to compare the typical or central value.
- A measure of spread: range, interquartile range (IQR), or standard deviation — to compare how consistent or varied the data is.
Which statistics to use when comparing
| Situation | Preferred average | Preferred spread |
|---|---|---|
| Data with outliers (extreme values) | Median | Interquartile range (IQR) |
| Symmetric data, no outliers | Mean | Range or IQR |
| Categorical/frequency data | Mode | — |
| Box plot given | Median | IQR |
| Cumulative frequency given | Median | IQR |
| Raw list of values | Mean or median | Range or IQR |
How do you compare two box plots?
Box plots show five values: minimum, lower quartile (Q1), median, upper quartile (Q3), and maximum. The box spans Q1 to Q3 (the IQR), and the line inside shows the median.
To compare two box plots:
Step 1 — Compare the medians. State which is larger and what this means in context.
Step 2 — Compare the IQRs. State which is larger and what this means about consistency.
Step 3 — Give a conclusion relevant to the question (e.g. "Group A performed better on average but was less consistent").
Worked example 1: comparing two exam score box plots
Group A: median 62, IQR = 75 − 50 = 25.
Group B: median 58, IQR = 70 − 42 = 28.
Comparison:
"Group A had a higher median score (62 compared to 58), suggesting Group A performed better on average. Group A also had a smaller IQR (25 compared to 28), indicating Group A's scores were more consistent."
This is a full GCSE mark-scheme answer: two comparative statements, each with evidence (numbers) and a contextual interpretation.
How do you compare distributions from cumulative frequency graphs?
Read the median from the 50th percentile and the IQR from the 25th and 75th percentiles.
Reading off a cumulative frequency graph:
- Total frequency = n.
- Lower quartile Q1: read at n/4.
- Median Q2: read at n/2.
- Upper quartile Q3: read at 3n/4.
- IQR = Q3 − Q1.
Then compare using the same two-statement structure as for box plots.
How do you compare distributions from frequency tables?
Calculate the mean from each frequency table (∑fx ÷ ∑f) and the range (or IQR if quartile positions are given). Use the calculated values as evidence for your comparison.
Worked example 2: comparing waiting times
| Waiting time (min) | Frequency (Hospital A) | Frequency (Hospital B) |
|---|---|---|
| 0 – 10 | 5 | 12 |
| 10 – 20 | 20 | 18 |
| 20 – 30 | 15 | 8 |
| 30 – 40 | 10 | 2 |
| Total | 50 | 40 |
Using midpoints for each class (5, 15, 25, 35):
Hospital A mean ≈ (5×5 + 15×20 + 25×15 + 35×10) ÷ 50 = (25 + 300 + 375 + 350) ÷ 50 = 1050 ÷ 50 = 21 min
Hospital B mean ≈ (5×12 + 15×18 + 25×8 + 35×2) ÷ 40 = (60 + 270 + 200 + 70) ÷ 40 = 600 ÷ 40 = 15 min
Comparison: "Hospital B had a lower mean waiting time (15 minutes compared to 21 minutes), suggesting shorter average waits. Hospital A has more patients in the 30–40 minute band, indicating greater variation in waiting times."
What phrases score marks in a comparison question?
Use these structures in your written answer:
- "The median for [Group A] is [value], which is higher/lower than the median for [Group B] of [value]. This suggests [Group A] [interpretation]."
- "The IQR for [Group A] is [value], which is smaller/larger than the IQR for [Group B] of [value]. This means [Group A]'s data is more/less consistent/varied."
Every comparison must include: the statistic name, both numerical values, a comparative word (higher/lower/greater/smaller), and a contextual interpretation.
Frequently asked questions
How many marks does a comparison question usually carry?
Most GCSE comparison questions carry 2 to 4 marks. A 2-mark question typically needs one comparison (e.g. medians only). A 4-mark question expects both an average comparison and a spread comparison, each with numerical evidence. Partial credit is common — one good comparison usually earns at least 1 mark.
Can I use the range instead of the IQR?
Yes, but the IQR is more reliable when outliers are present, because the range is determined by the extreme values. On a box plot, IQR is directly visible; for raw data, the range is quick to calculate. State which measure you are using: "The range for Group A is 42, compared to 38 for Group B."
Do I need to say which group is "better"?
Yes, if the question includes context (e.g. "two runners' race times", "two classes' exam scores"). Interpret your statistics in context: a lower median for race times means faster; a higher median for exam scores means better performance. Stating only the numbers without interpretation misses the communication marks.
What if the two data sets have different sample sizes?
The averages (mean, median) are still comparable directly. However, comparing raw totals or frequencies is meaningless without standardising. If you are comparing cumulative frequency graphs with different totals, read the quartile positions as fractions of n (i.e. n/4 and 3n/4) rather than at fixed values — that way both graphs are on the same relative scale.
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