Cumulative Frequency and Box Plots –

Cumulative frequency diagrams and box plots are Higher-tier statistics topics that appear on AQA, Edexcel and OCR GCSE papers almost every year. They allow you to estimate the median, quartiles and interquartile range from grouped data — all things you cannot find exactly from a grouped frequency table alone. Being able to draw and interpret both representations confidently is worth several marks in the exam. This guide takes you through the method step by step, provides worked examples with full solutions, flags the traps examiners set, and gives you practice questions to test yourself. See our complete GCSE Maths topics list for context on where this fits.

What Is Cumulative Frequency?

Cumulative frequency is a running total of frequencies. For each class interval, you add the frequency of that class to the total of all previous classes. The result tells you how many data items are less than or equal to the upper boundary of that class.

Cumulative Frequency Diagram

A cumulative frequency diagram (sometimes called an ogive) plots the upper class boundary on the horizontal axis against the cumulative frequency on the vertical axis. The points are joined with a smooth S-shaped curve (or straight-line segments). You can then read off estimates for the median, lower quartile (LQ) and upper quartile (UQ).

Box Plot (Box-and-Whisker Diagram)

A box plot is a five-number summary of data displayed on a number line:

1. Minimum value
2. Lower quartile (Q1) — 25% of the data lies below this
3. Median (Q2) — 50% of the data lies below this
4. Upper quartile (Q3) — 75% of the data lies below this
5. Maximum value

The "box" spans from Q1 to Q3. A line inside the box marks the median. "Whiskers" extend from the box to the minimum and maximum values.

Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of the data. It is not affected by outliers and is therefore a more reliable measure of spread than the range.

Step-by-Step Method

Drawing a Cumulative Frequency Diagram

1. Add a cumulative frequency column to your grouped frequency table — keep a running total.
2. Plot each point at the upper class boundary (not the midpoint) against the cumulative frequency.
3. Plot a point at the lower boundary of the first class with a cumulative frequency of 0.
4. Join the points with a smooth curve.
5. Label both axes: horizontal with the data variable and units, vertical with "Cumulative frequency".

Reading the Median and Quartiles

1. Find n (the total frequency).
2. For the median, go across from n ÷ 2 on the vertical axis to the curve, then read down.
3. For Q1, use n ÷ 4.
4. For Q3, use 3n ÷ 4.
5. Calculate the IQR = Q3 − Q1.

Drawing a Box Plot

1. Draw a number line covering the range of the data.
2. Mark the minimum, Q1, median, Q3 and maximum with short vertical lines.
3. Draw a box from Q1 to Q3.
4. Draw a vertical line inside the box at the median.
5. Draw whiskers from the box to the minimum and maximum.

Worked Example 1 — Higher Level

Question: The table shows the distances (km) 80 people travel to work.

(a) Draw a cumulative frequency diagram. (b) Use your diagram to estimate the median, Q1, Q3 and the IQR.

Working:

(a) Cumulative frequency table:

Plot (0, 0), (5, 8), (10, 28), (15, 56), (20, 72), (30, 80) and join with a smooth curve.

(b) Estimates:

• Median: n ÷ 2 = 40. Read across from 40 on the CF axis to the curve, then down: approximately 12.5 km.
• Q1: n ÷ 4 = 20. Read across from 20: approximately 8.5 km.
• Q3: 3n ÷ 4 = 60. Read across from 60: approximately 17 km.
• IQR = 17 − 8.5 = 8.5 km.

Worked Example 2 — Higher Level

Question: Two year groups take the same test. Their results are summarised by box plots.

• Year 10: Min = 22, Q1 = 40, Median = 55, Q3 = 68, Max = 90
• Year 11: Min = 30, Q1 = 48, Median = 62, Q3 = 74, Max = 95

Compare the two distributions.

Working:

Comparing averages: Year 11 has a higher median (62 vs 55), suggesting Year 11 performed better on average.

Comparing spread: Year 11 IQR = 74 − 48 = 26. Year 10 IQR = 68 − 40 = 28. The IQR is slightly smaller for Year 11, meaning their middle 50% of marks were more consistent.

Comparing range: Year 10 range = 90 − 22 = 68. Year 11 range = 95 − 30 = 65. The ranges are similar.

Conclusion: Year 11 generally scored higher and with slightly more consistency than Year 10.

Common Mistakes

• Plotting at midpoints instead of upper class boundaries — cumulative frequency points must be plotted at the upper boundary.
• Using n ÷ 2 + 1 instead of n ÷ 2 — for grouped data and cumulative frequency curves, use n ÷ 2 (the +1 is for listed data only).
• Drawing a jagged line instead of a smooth curve — the examiner expects a smooth S-shaped curve through the points.
• Forgetting the (0, 0) point — start the curve at the lower boundary of the first class with a cumulative frequency of zero.
• Confusing IQR with range — the IQR is Q3 − Q1, not max − min.
• Not comparing like with like — when comparing box plots, comment on both an average (median) and a measure of spread (IQR or range).

Exam Tips

1. Draw accurately — use the full width of the graph paper and plot points with small crosses, not dots.
2. Read values carefully — when estimating from the curve, draw horizontal and vertical guide lines with a ruler and pencil. The examiner will check your reading to within half a square.
3. Comparison questions — always make two comparative statements: one about an average (median) and one about spread (IQR). Use the context of the question (e.g. "Year 11 performed better on average because…").
4. Box plots can appear alongside histograms — draw them on the same horizontal scale if required. For histogram basics, see histograms.
5. Revisit key formulas on our GCSE Maths formulas page.

Practice Questions

Question 1: The cumulative frequency table for the heights of 50 plants is shown.

Estimate the median and the IQR.

Question 2: Draw a box plot given: Min = 12, Q1 = 18, Median = 25, Q3 = 33, Max = 42.

Question 3: Two data sets have box plots. Set A: median = 45, IQR = 20. Set B: median = 52, IQR = 12. Compare the two distributions.

Question 4: Explain why the interquartile range is often preferred to the range as a measure of spread.

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Related Topics

• Histograms
• Mean, Median, Mode and Range
• Frequency Tables and Grouped Data
• Scatter Graphs and Correlation

Summary

Cumulative frequency diagrams and box plots let you estimate the median, quartiles and interquartile range from grouped data. To build a cumulative frequency curve, calculate a running total of frequencies and plot each value against the upper class boundary, then join the points with a smooth S-curve. Read off the median at n ÷ 2, Q1 at n ÷ 4 and Q3 at 3n ÷ 4. A box plot summarises data using five values — minimum, Q1, median, Q3 and maximum — and is excellent for comparing two distributions. When comparing, always comment on an average (median) and a measure of spread (IQR). These topics link tightly to histograms, grouped frequency tables and calculating averages.