Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Cumulative Frequency & Box Plots

In one sentence. Running totals, medians and quartiles read from the curve, box plots, and comparing distributions properly. Higher Tier.

Statistics — one of 3 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Adolphe Quetelet. Learn Cumulative Frequency & Box Plots, teach it, then defend it against Adolphe Quetelet.

Where this sits in the specification

Specification 4MA1, 6.1 Cumulative frequency, quartiles, box plots (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 6.1 — Cumulative frequency, quartiles, box plots (Higher)

1 · Understand it

What it is

Cumulative frequency is the running total of data 'so far'; plotted against the upper class boundaries it makes an S-curve. Reading positions: for n data points, the median sits at n/2, the lower quartile at n/4, the upper at 3n/4 — read ACROSS to the curve, then down. IQR = UQ − LQ. A box plot draws minimum, LQ, median, UQ, maximum in one strip.

Why it works

The curve answers 'how many are at or below this value?' — so reading it backwards at the halfway count finds the value the middle person has: the median. The IQR spans the middle 50% — a spread measure that outliers can't touch, which is exactly why it partners the median.

When a question is asking for it

'Use the graph to estimate the median/quartiles/IQR'; 'how many scored more than…?' (read the curve, subtract from n); and 'compare the distributions' — which ALWAYS wants one average compared AND one spread compared, in context.

How to hold it in your head

Imagine the data lined up shortest to tallest. The median is person n/2's value; quartiles are the quarter- and three-quarter-way people. The CF curve is just a machine for looking those people up without doing the lining-up yourself.

2 · The mistakes that cost marks

  • Plotting CF against class MIDpoints. The running total is complete only at the UPPER boundary.
  • Reading the median at half the largest VALUE instead of half the total FREQUENCY.
  • IQR quoted as the pair 'LQ to UQ' rather than the difference UQ − LQ.
  • Comparing two data sets by medians alone. A full comparison pairs an average WITH a spread, both interpreted.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A cumulative frequency graph shows 60 values. At what cumulative frequency do you read off the median, and the two quartiles?

  1. Median: n/2 = 30th position.
  2. LQ: n/4 = 15th. UQ: 3n/4 = 45th.
  3. Read across from those counts to the curve, then down to the values.

Answer: 30 (median), 15 (LQ), 45 (UQ)

From a CF graph of 60 test scores, 42 students scored 50 or less. How many scored MORE than 50?

  1. The curve says 42 are at or below 50.
  2. 60 − 42 = 18 above.

Answer: 18

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — A cumulative frequency graph shows 60 values. At what cumulative frequency (position) should you read off the MEDIAN?
  2. Grade 7 — On the same graph of 60 values, the curve gives a value of 34 at cumulative frequency 15, and 58 at cumulative frequency 45. Work out the interquartile range.
  3. Grade 8 — A box plot shows: minimum 12, lower quartile 20, median 26, upper quartile 34, maximum 50. Work out the interquartile range.
  4. Grade 9 — A cumulative frequency graph shows the finishing times of 80 runners. Reading from it, 26 runners took 40 minutes or less and 71 runners took 55 minutes or less. What percentage of the runners took MORE than 40 minutes but no more than 55 minutes? (answer in %)
  5. 9+ — For a data set of 60 values shown on a cumulative frequency graph, at what cumulative frequency should you read off the 90th percentile?

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Cumulative Frequency & Box Plots, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Cumulative Frequency & Box Plots works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Cumulative Frequency & Box Plots against Adolphe Quetelet — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

Completely free. No subscriptions, no paywalls, no account needed to start.

Carry on through the course