Cumulative frequency and box plots
Medians, quartiles and interquartile range from a curve.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 9.2.1Draw and interpret cumulative frequency diagrams and box-and-whisker plots.
6 minute read
Cumulative frequency and box plots
Cumulative frequency
Cumulative frequency is a running total. Each value is added to all those before it, so the final figure equals the total number of items. Plot cumulative frequency against the upper class boundary of each group — not the midpoint — and join with a smooth curve. Plotting at the midpoint is a standard error.
Reading the curve
For n items:
- Median — read across from
n/2. - Lower quartile (Q1) — read across from
n/4. - Upper quartile (Q3) — read across from
3n/4. - Interquartile range (IQR) =
Q3 − Q1.
Note that for a curve we use n/2 rather than (n+1)/2 — the smoothed graph treats the data as continuous.
Why the IQR matters
The IQR measures the spread of the middle half of the data, so it ignores outliers entirely. That is precisely why it is often preferred to the range, which is defined by the two most extreme values.
Box-and-whisker plots
A box plot displays five values: minimum, Q1, median, Q3, maximum. The box spans the IQR with the median marked inside; the whiskers reach the extremes.
Comparing two distributions
Always compare two things:
- An average — usually the medians — for typical value.
- A spread — usually the IQRs — for consistency.
"Group A has a higher median so scored better on average, and a smaller IQR so was more consistent." That two-part structure is what the marks are for.
Think of it like this
A box plot is a five-number summary of a whole dataset — like a weather summary giving the low, high and typical temperature rather than every hourly reading.
Worked examples
Method, step by step
A cumulative frequency curve represents 80 students. Explain how to find the interquartile range from it.
- 1Q1 is at n/4 = 80/4 = 20; read across from 20 to the curve and down to the value.
- 2Q3 is at 3n/4 = 60; read across from 60 to the curve and down.
- 3IQR = Q3 − Q1.
Read the values at cumulative frequencies 20 and 60, then subtract the lower from the upper.
Common misconceptions
- Plotting cumulative frequency against the midpoint of a class. It must be the **upper** boundary.
- Comparing only the medians. Spread must also be commented on for full marks.
- Confusing the IQR with the range. The IQR is Q3 − Q1 and deliberately excludes the extremes.
In the exam
- Draw and label your reading lines on the graph — examiners award marks for the correct construction lines even if the final read-off is slightly out.
- Structure comparisons in two sentences: one about average, one about spread, each interpreted in context.