Key Stage 4 maths · Statistics

GCSE histograms, cumulative frequency and box plots (higher)

The higher-tier GCSE adds more powerful ways to display and compare data. Children construct and interpret histograms, including those with unequal class widths, draw cumulative frequency graphs to find the median and quartiles, and use box plots and the inter-quartile range to compare how two data sets are spread. They also learn which display is appropriate for which kind of data.

Histograms and box plots, explained

A histogram looks like a bar chart but works differently. It is used for grouped continuous data (and sometimes grouped discrete data), and the bars have no gaps. When class widths are unequal, the height of a bar can’t be the frequency, because a wide class would look unfairly large. Instead the vertical axis shows frequency density, which is frequency ÷ class width, so that the area of each bar represents the frequency. This is why a histogram can be used to estimate how many values lie in part of a class: find the area of that part, assuming the data is spread evenly.

A cumulative frequency graph shows a running total. A table is made with the cumulative frequency for each class, and the points are plotted at the upper boundary of each class, since that is the point by which all the values in the class have been counted. They are then usually joined with a smooth curve. From the graph children read the median at half the total frequency, the lower quartile at a quarter and the upper quartile at three quarters. They can also estimate how many values are below a given number.

Quartiles split ordered data into four equal parts. The lower quartile (Q1) has a quarter of the data below it, the median (Q2) has half and the upper quartile (Q3) has three quarters. The inter-quartile range, Q3 − Q1, measures the spread of the middle half of the data and is hardly affected by a few extreme values. For the nine values 3, 5, 6, 8, 9, 11, 12, 14, 15, the median is 9, Q1 is 5.5, Q3 is 13 and the inter-quartile range is 7.5. A box plot draws the five key values, the lowest, Q1, median, Q3 and highest, as a box with two whiskers.

Because each section of a box plot contains about a quarter of the data, box plots are excellent for comparing two groups on the same scale. A good comparison mentions the medians (the typical value) and the inter-quartile ranges (the spread) in the context of the question. Children also need to know the right tool for the job: histograms for grouped continuous data, cumulative frequency for finding medians and quartiles from grouped data, and box plots for comparing distributions.

Words your child will hear

Frequency density: Frequency divided by class width. It is the height of a bar in a histogram.
Class width: The size of a group, found by subtracting its lower boundary from its upper boundary.
Cumulative frequency: A running total of frequencies, adding each class to all those before it.
Quartiles: The values that split ordered data into four equal parts: lower quartile, median and upper quartile.
Inter-quartile range: The upper quartile minus the lower quartile: the spread of the middle half of the data.
Box plot: A diagram showing the lowest value, lower quartile, median, upper quartile and highest value.
Continuous data: Data that can take any value in a range, such as time or height, instead of separate values.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals, and know their appropriate use
Construct and interpret cumulative frequency graphs for grouped discrete data and continuous data, and know their appropriate use
Interpret, analyse and compare distributions of data sets using box plots
Interpret, analyse and compare distributions of data sets using measures of spread including quartiles and inter-quartile range

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

The table shows how long 60 customers waited to be served. Work out the frequency density for each class, then estimate how many customers waited between 20 and 30 minutes.

Frequency density = frequency ÷ class width. Add a column for the class width and one for the frequency density.

Wait t (min)     f  Width    FD
0 ≤ t < 10     12     10   1.2
10 ≤ t < 20    18     10   1.8
20 ≤ t < 40    20     20   1.0
40 ≤ t < 60    10     20   0.5

These are the bar heights for the histogram. Notice that the 20 to 40 class has more customers (20) than the 10 to 20 class (18), yet its bar is lower. That is because the class is twice as wide, so the same number of people is spread more thinly.
The area of a bar equals its frequency, because height × width = frequency ÷ width × width. So 1.2 × 10 = 12, 1.8 × 10 = 18, 1.0 × 20 = 20 and 0.5 × 20 = 10. These add to 60, which matches the total.
For 20 to 30 minutes, take part of the 20 to 40 bar. That portion is 10 minutes wide and 1.0 high, so its area is 1.0 × 10 = 10. This assumes the customers are spread evenly through the class.

The frequency densities are 1.2, 1.8, 1.0 and 0.5. About 10 customers waited between 20 and 30 minutes.

Common mistakes and how to help

Using frequency as the height of the bars when class widths are unequal

In bar charts and equal-width histograms, height does show frequency, so the habit transfers. With unequal widths, wide classes look far too big.

What helps: Say “area, not height, shows frequency”. Always work out the frequency density first, and check that the total of all the bar areas equals the total frequency.

Plotting cumulative frequency at the class midpoint or the lower boundary

Midpoints are used for estimating the mean, so they come to mind. But a running total is only complete at the end of a class.

What helps: Plot each cumulative frequency against the upper class boundary, starting the graph at the lowest boundary with a cumulative frequency of 0, and join the points with a smooth curve.

Thinking a longer box or whisker means more data

In a box plot a longer section looks as if it holds more values. Each section holds roughly a quarter of the data, so a long section just means the values there are more spread out.

What helps: Count the sections: four quarters of the data, whatever their lengths. A short box means the middle half is tightly packed, and a long one means it is widely spread.

Comparing groups with numbers but no sentence in context

An answer like “A has a median of 42 and B has 38” states values without saying what they mean, and does not use any measure of spread.

What helps: Use a pattern: one comparison of averages and one of spread, each tied to the situation. “The median wait at A is higher, so customers typically wait longer, and the inter-quartile range is smaller, so waits there are more consistent.”

How to help at home

Ruler-drop reaction test. One person holds a 30 cm ruler and lets go without warning, and the other catches it. Record the distance fallen. Collect 20 tries each, then find the median and quartiles for each person. A shorter distance means a quicker reaction. Draw the two box plots on the same scale and write two sentences comparing them.
Quartiles from number cards. Write 11 numbers on cards and lay them out in order. The 6th card is the median, the 3rd is the lower quartile and the 9th is the upper quartile. Then draw a box plot beneath them on squared paper, using one square per unit.
Compare unequal groups. List the ages of family and friends in groups such as 0 up to 10, 10 up to 20, 20 up to 60 and 60 up to 100. Draw a bar chart of the frequencies and then a histogram using frequency density. Discuss which picture is more honest.
Build a cumulative frequency graph. Note the length of 30 songs, or 30 journey times over a few weeks. Group them, make a cumulative frequency table, plot it at the upper class boundaries and read off the median and quartiles. Then draw a box plot from those values and ask your child to explain what the box and whiskers show.

Builds on

Key Stage 3: Averages, charts and scatter graphs