Cumulative Frequency Graphs & Box Plots — AQA GCSE Mathematics

SPEC S3 + S4(H) • HIGHER TIER ONLY

Cumulative Frequency & Box Plots

Graphs that reveal the median, the quartiles and the spread of grouped data. Higher-tier only — high-priority on every Higher paper.

Why Cumulative Frequency? Higher Tier Only

Estimated mean tells you a typical value — but it doesn't tell you where the middle sits, or how spread out the data is. Cumulative frequency graphs fix both for grouped data.

What you'll learn (all Higher-tier content)

Build a cumulative frequency table and plot a cumulative frequency graph.

Read the median, lower and upper quartiles, and percentiles from a CF graph.

Calculate the interquartile range (IQR).

Draw and interpret a box plot from a five-number summary.

Compare two distributions using an average and a measure of spread, both in context.

MCE approach

Method — plot CF against the upper bound, read across from the CF-axis to the curve, drop down to the data-axis. Calculate — median at n/2, quartiles at n/4 and 3n/4, IQR = Q3 − Q1. Evaluate — check readings are within the data range and comparisons are in context.

What is Cumulative Frequency? Higher Tier Only

Cumulative frequency

The running total of frequencies up to and including that class. At the upper bound of each class, CF = the number of data values at or below that bound.

Example: 50 students reported their weekly homework time.

Time t (min)FrequencyCumulative frequencyMeaning

0 ≤ t < 10555 students did less than 10 min

10 ≤ t < 20155 + 15 = 2020 students did less than 20 min

20 ≤ t < 302020 + 20 = 4040 students did less than 30 min

30 ≤ t < 40840 + 8 = 4848 students did less than 40 min

40 ≤ t < 50248 + 2 = 50All 50 students did less than 50 min

Two sanity checks

CF only ever increases (or stays the same).

The final CF value must equal n, the total number of data values.

Plotting the Cumulative Frequency Graph Higher Tier Only

Three plotting rules

Plot CF against the upper bound of each class — not the midpoint.

Include the point (lower bound of first class, 0) to anchor the curve at zero.

Join the points with a smooth curve, not straight lines.

Points: (0, 0), (10, 5), (20, 20), (30, 40), (40, 48), (50, 50) — joined with a smooth curve.

Interpreting the Curve Higher Tier Only

A cumulative frequency curve is usually an S-shape for grouped data because the CF accumulates more rapidly where there are more data values. The exact shape depends on how the data is distributed.

What the shape tells you

A steep section means many data values fall in that class — high frequency.

A flat section means few data values there — low frequency.

The curve starts at CF = 0 (lower bound of first class) and levels off at CF = n.

Criticising a CF graph

AQA often shows a slightly-wrong CF graph and asks students to criticise it. Common correct criticisms:

"Points plotted at midpoint, not upper bound."

"Straight lines instead of a smooth curve."

"Axis labelled 'frequency' instead of 'cumulative frequency'."

"One of the points is plotted at the wrong height."

Reading the Median Higher Tier Only

For grouped data, the median is the value at the n/2 position on the CF-axis.

Median from a CF graph

Find n/2 on the cumulative frequency (vertical) axis.

Draw a horizontal line from that value across to the curve.

Drop a vertical line down to the horizontal axis.

Read the value — that's the median.

Median ≈ 22 min — horizontal from CF = 25 to the curve, then drop down. In an exam, readings within ±0.5 small squares of the correct value earn full marks.

Worked Example — Build CF Table & Find Median

Higher Tier Only

The table shows the masses of 40 parcels at a depot. (a) Complete the cumulative frequency column. (b) Use a CF graph to estimate the median mass. [1 + 3 marks]

Mass m (kg)FrequencyCumulative frequency

0 ≤ m < 26?

2 ≤ m < 414?

4 ≤ m < 612?

6 ≤ m < 86?

8 ≤ m < 102?

MMethod. (a) Running totals of the frequencies — the final total must equal 40. (b) Median at CF = n/2 = 20. Plot CF against upper bounds, join with a smooth curve, read across from CF = 20.

Calculate.

(a) Running totals: 6, 6 + 14 = 20, 20 + 12 = 32, 32 + 6 = 38, 38 + 2 = 40 ✓.

Plot points: (0, 0), (2, 6), (4, 20), (6, 32), (8, 38), (10, 40).

(b) n/2 = 20. CF = 20 is exactly the CF value at the point (4, 20), so the horizontal meets the curve at m = 4. Median ≈ 4 kg.

EEvaluate. Median = 4 kg sits at the boundary of the two central classes (2 ≤ m < 4 and 4 ≤ m < 6) — the 14 + 12 = 26 parcels in these two classes are the bulk of the data, which is consistent.

Quartiles Higher Tier Only

Quartiles split the ordered data into four equal groups of size n/4. They're read off a CF graph exactly like the median — just at different CF values.

Lower and upper quartile

Lower quartile (Q1) — the value at CF = n/4. One quarter of the data lies below it.

Upper quartile (Q3) — the value at CF = 3n/4. Three quarters of the data lies below it.

Median (Q2) — the value at CF = n/2. Half lies below.

Reading off: the three horizontal lines

For n = 80: Q1 at CF = 20, median at CF = 40, Q3 at CF = 60. Three horizontal lines → three readings on the data axis.

Percentiles work the same way

The p-th percentile is the value at CF = (p / 100) × n. So the 80th percentile on 200 data values is at CF = 160. The median is the 50th percentile; Q1 is the 25th; Q3 is the 75th.

Interquartile Range Higher Tier Only

Interquartile range (IQR)

IQR = Q3 − Q1The IQR measures the spread of the middle 50% of the data. It ignores the extremes, so it's a robust measure of spread — not affected by outliers.

IQR vs range

Range = max − min. Sensitive to outliers; one extreme value inflates it.

IQR = Q3 − Q1. Robust — only the middle 50% of the data affects it.

A smaller IQR means the middle 50% of the data clusters tightly — the data is more consistent.

Sign matters: IQR is always positive

Always compute Q3 − Q1, not Q1 − Q3. The IQR is a spread, so it must be positive.

Worked Example — Quartiles from a CF Graph

Higher Tier Only

80 students sat a test scored out of 100. From the CF graph, estimate (a) the median, (b) the lower and upper quartiles, (c) the interquartile range. [4 marks]

MMethod. n = 80. Q1 at CF = n/4 = 20. Median at CF = n/2 = 40. Q3 at CF = 3n/4 = 60. Draw horizontal lines across from the CF axis, then vertical lines down to the data axis.

Calculate.

•

(a) Median: at CF = 40 → score = 60 marks .

•

(b) Q1: at CF = 20 → score = 40 marks . Q3: at CF = 60 → score = 80 marks .

•

(c) IQR = Q3 − Q1 = 80 − 40 = 40 marks .

EEvaluate. Q1 < Median < Q3 (40 < 60 < 80) ✓. IQR is positive. All three readings lie within the data range (0 to 100) ✓.

Box Plots Higher Tier Only

A box plot summarises a distribution using just five numbers, drawn on a number line.

Five-number summary

Minimum · Lower quartile · Median · Upper quartile · Maximum

What each part shows

The box contains the middle 50% of the data (from Q1 to Q3). Its width is the IQR.

The median line sits at the exact median value — not the centre of the box.

The whiskers extend to the minimum and maximum values. Total length = the range.

Drawing a Box Plot Higher Tier Only

Six-step routine

Draw a number line covering the full data range.

Mark the five values: min, Q1, median, Q3, max.

Draw a box from Q1 to Q3.

Draw a vertical line inside the box at the median — not the centre.

Draw whiskers from the box edges to the min and max.

Add short vertical caps at the whisker ends.

Two common plotting errors

Median line at the centre of the box — should be at the exact median value. The box usually looks uneven when drawn correctly.

Whiskers drawn to Q1 and Q3 — should extend beyond the box to the min and max.

Reverse direction: reading a box plot

Sometimes a box plot is given and you're asked to find the median, IQR, or range. Just read off the five positions: left edge of the box is Q1, inner line is the median, right edge is Q3, whisker caps are min and max. IQR = Q3 − Q1, range = max − min.

Comparing Distributions Higher Tier Only

Two box plots on the same axis let you compare two datasets at a glance. A full-mark comparison needs two statements, both in context.

The two required comparisons

An average: compare the medians (or means if given). Say which is higher/lower, in the context of the data.

A spread: compare the IQRs (or ranges). Say which dataset is more consistent or more spread out, in the context of the data.

Context is what scores the mark

AQA mark schemes consistently reject non-contextual statements. For battery life, for example:

❌ "P's median is higher" — no context (B0).

✅ "Brand P lasts longer because its median is 132 hours vs 126 hours" — context (B1).

❌ "P's IQR is smaller" — no context (B0).

✅ "Brand P is more consistent because its IQR is 47 vs 57" — context (B1).

Useful contextual vocabulary

"Lasts longer / scored higher / grew taller" (average); "more consistent / more reliable / more varied / less predictable" (spread).

Worked Example — Compare Two Box Plots

Higher Tier Only

The box plots show the test scores for two classes, A and B. Compare the distributions. [2 marks]

MMethod. Read off the median and IQR for each class. Make one statement about averages (in context) and one about spread (in context).

Calculate.

•

Class A: median = 55 , Q1 = 45, Q3 = 65, IQR = 65 − 45 = 20 .

•

Class B: median = 50 , Q1 = 35, Q3 = 70, IQR = 70 − 35 = 35 .

Evaluate — the two comparison statements.

Average: "Class A scored higher on average than Class B because A's median (55) is greater than B's median (50)."

Spread: "Class A's scores were more consistent than Class B's because A's IQR (20) is smaller than B's IQR (35)."

Examiner Alerts

Alert 1 — Plotting at the midpoint instead of the upper bound

The cumulative total is reached at the upper bound of each class, so plot CF at the upper bound — not the midpoint. This is the single most common CF error.

Alert 2 — Straight lines instead of a smooth curve

Connect the points with a smooth curve, not straight line segments.

Alert 3 — IQR sign error

IQR = Q3 − Q1, always. Writing Q1 − Q3 gives a negative number, which cannot be a spread.

Alert 4 — Non-contextual comparisons

"A has a higher median" and "A has smaller IQR" are both B0. Rewrite in context: "Brand A lasts longer on average" and "Brand A is more consistent". Two statements, both in context.

Alert 5 — Box plot median at the centre of the box

The median line inside the box goes at the median value, which is usually not the centre of the box. A correctly drawn box plot is normally asymmetric.

Practice Question Higher Tier Only

A grower records the weights of 40 apples. The cumulative frequency table is shown below.

Weight w (g)FrequencyCumulative frequency

100 ≤ w < 12044

120 ≤ w < 1401014

140 ≤ w < 1601226

160 ≤ w < 1801036

180 ≤ w < 200440

(a) Use a CF graph to estimate the median weight. [2 marks]

(b) Estimate the lower quartile, upper quartile and the interquartile range. [3 marks]

(c) The minimum weight is 100 g and the maximum is 200 g. Draw a box plot for the data. [2 marks]

MMethod. n = 40. Median at CF = 20. Q1 at CF = 10. Q3 at CF = 30. Box plot uses (min, Q1, median, Q3, max).

Calculate.

(a) CF = 20 lies between (140, 14) and (160, 26). Moving 6 of 12 = half way, so median ≈ 140 + 0.5 × 20 = 150 g.

(b) CF = 10 lies between (120, 4) and (140, 14). 6 of 10 = 0.6, so Q1 ≈ 120 + 0.6 × 20 = 132 g. CF = 30 lies between (160, 26) and (180, 36). 4 of 10 = 0.4, so Q3 ≈ 160 + 0.4 × 20 = 168 g. IQR = 168 − 132 = 36 g.

(c) Box plot: min = 100, Q1 = 132, median = 150, Q3 = 168, max = 200. Box from 132 to 168, median line at 150, whiskers to 100 and 200.

EEvaluate. Q1 (132) < median (150) < Q3 (168), IQR is positive, all values within 100–200 g range ✓. The median line in the box plot sits slightly left of centre, consistent with the data being slightly skewed towards lower weights.

Summary — MCE Checkpoint Higher Tier Only

Cumulative Frequency (S3, Higher only)

CF = running total of frequencies. Last value = n.

Plot CF against the upper bound of each class.

Include the point (lower bound of first class, 0).

Join with a smooth curve .

Median at CF = n/2 — read across, then down.

Quartiles, IQR, Box Plots (S4, Higher only)

Q1 at CF = n/4, Q3 at CF = 3n/4.

IQR = Q3 − Q1 — middle 50% spread, robust to outliers.

Box plot : five numbers — min, Q1, median, Q3, max. Box from Q1 to Q3, median line at the median (not centre).

Percentile: p-th percentile at CF = (p/100) × n.

Comparing distributions

Need two statements: one average, one spread.

Both must be in context — "lasts longer" not "has higher median".

Smaller IQR or range = more consistent .

Examiner traps

Plot at upper bound , not midpoint.

Use a smooth curve , not straight lines.

IQR = Q3 − Q1 — always positive.

Median line at the median value, not the centre of the box.

Comparisons must be in context .

✓ You should now be able to

Construct a cumulative frequency table from a grouped frequency table.

Plot a cumulative frequency graph with correct conventions (upper bound, smooth curve).

Read the median, lower and upper quartiles from a CF graph.

Calculate the IQR and use it as a robust measure of spread.

Draw and interpret a box plot from a five-number summary.

Compare two distributions with contextual statements about average and spread.

Identify and correct common criticisms of flawed CF graphs.