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Cumulative frequency, box plots and histograms GCSE questions with answers

8 cumulative frequency, box plots and histograms questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

On Higher, this topic covers cumulative frequency graphs, box plots and histograms. You read medians and quartiles from a cumulative frequency graph, compare data with box plots and use frequency density in histograms.

Quiz yourself: 8 questions

Worked example 1: box plots

The box plot shows the marks some pupils got in a test. Write down the median.

0102030405060Mark
  1. Reading the box plot: lowest 16, lower quartile 34, median 42, upper quartile 46, highest 50. Each small mark on the scale is 2.
  2. The median is the line inside the box: 42.

Answer 42

01 / Quiz yourselfTry it, then tap to check

Cumulative frequency, box plots and histograms questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Box plotsHigher only

The box plot shows the marks some pupils got in a test. Write down the range.

0102030405060Mark
Show the answer and working
  1. Reading the box plot: lowest 14, lower quartile 24, median 28, upper quartile 44, highest 56. Each small mark on the scale is 2.
  2. Range = highest − lowest = 56 − 14 = 42.

Answer 42

02 / Box plotsHigher only

The box plot shows the marks some pupils got in a test. Write down the median.

0102030405060Mark
Show the answer and working
  1. Reading the box plot: lowest 8, lower quartile 16, median 22, upper quartile 34, highest 44. Each small mark on the scale is 2.
  2. The median is the line inside the box: 22.

Answer 22

03 / Box plotsHigher only

The box plot shows the times, in minutes, that some pupils spent on homework one evening. Write down the interquartile range.

020406080100120Time (minutes)
Show the answer and working
  1. Reading the box plot: lowest 10, lower quartile 25, median 60, upper quartile 70, highest 80. Each small mark on the scale is 5.
  2. Interquartile range = upper quartile − lower quartile = 70 − 25 = 45.

Answer 45

04 / Cumulative frequency graphsHigher only

The cumulative frequency graph shows the masses, in grams, of 36 apples. Use the graph to estimate the median.

80100120140160180010203040Mass (g)Cumulative frequency
Show the answer and working
  1. The median is halfway through the data: 36 ÷ 2 = 18.
  2. Go across from 18 on the cumulative frequency axis to the curve, then straight down.
  3. The median is about 130. Answers from 126 to 134 are fine, because readings off a graph are never exact.

Answer 130

05 / Cumulative frequency graphsHigher only

The cumulative frequency graph shows the times, in minutes, that 84 runners took to finish a fun run. Use the graph to estimate the median.

20304050607080020406080100Time (minutes)Cumulative frequency
Show the answer and working
  1. The median is halfway through the data: 84 ÷ 2 = 42.
  2. Go across from 42 on the cumulative frequency axis to the curve, then straight down.
  3. The median is about 49. Answers from 47 to 51 are fine, because readings off a graph are never exact.

Answer 49

06 / Cumulative frequency graphsHigher only

The cumulative frequency graph shows the marks of 84 pupils in a test. Use the graph to estimate the upper quartile.

0102030405060020406080100MarkCumulative frequency
Show the answer and working
  1. The upper quartile is three quarters of the way through the data: 3 × 84 ÷ 4 = 63.
  2. Go across from 63 on the cumulative frequency axis to the curve, then straight down.
  3. The upper quartile is about 38. Answers from 36 to 40 are fine, because readings off a graph are never exact.

Answer 38

07 / Histograms: frequency densityHigher only

The histogram shows the heights of some seedlings. How many seedlings are between 20 cm and 30 cm tall?

00.511.522.533.544.5Height (cm)Frequency density010152030
Show the answer and working
  1. Frequency = frequency density × class width (the area of the bar).
  2. The bar for 20 < h ≤ 30 has height 4 and width 30 − 20 = 10.
  3. 4 × 10 = 40.

Answer 40

08 / Histograms: frequency densityHigher only

The table shows how long some people waited at a doctor’s surgery. A histogram is to be drawn. Work out the frequency density for the class 30 < t ≤ 50.

Time (t minutes)0 < t ≤ 1010 < t ≤ 1515 < t ≤ 2020 < t ≤ 3030 < t ≤ 50Frequency161210856
Show the answer and working
  1. Frequency density = frequency ÷ class width.
  2. The class 30 < t ≤ 50 has width 50 − 30 = 20.
  3. 56 ÷ 20 = 2.8.

Answer 2.8

02 / What's covered3 kinds of question

Cumulative frequency, box plots and histograms at GCSE and where marks go

  • Box plotsHigher only
  • Cumulative frequency graphsHigher only
  • Histograms: frequency densityHigher only

Worked example 2: histograms: frequency density

The histogram shows how long some people waited at a doctor’s surgery. Estimate how many people waited between 25 and 35 minutes.

00.40.81.21.622.42.83.23.64Time (minutes)Frequency density0102025304060
  1. Frequency = frequency density × class width (the area of the bar).
  2. Where only part of a bar is in the range, use just that part of its width (assume the values are spread evenly).
  3. From 25 to 30: 0.8 × 5 = 4.
  4. From 30 to 35: 1.6 × 5 = 8.
  5. Add them: 12.

Answer 12

A common mistake

Reading the height of a histogram bar as the frequency. The height is frequency density, so frequency = frequency density × class width. A bar 2.5 high over a class width of 10 means a frequency of 25.

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