Histograms
A histogram is a chart for continuous grouped data with unequal class widths, where the area of each bar (not its height) represents the frequency of that class, using frequency density = frequency divided by class width on the vertical axis. It appears on GCSE Higher statistics papers, testing reading, drawing and estimating averages from histograms.
Before you start
Make sure you're comfortable with these topics first:
Method
- For each class, calculate frequency density using frequency density = frequency divided by class width.
- To find a missing frequency from a histogram, read off the bar height (frequency density) and multiply by the class width.
- When drawing a histogram, plot frequency density on the vertical axis against the class boundaries on the horizontal axis, with bars touching since the data is continuous.
- To estimate the number of items in part of a class, assume the data is evenly spread within that class and take the appropriate proportion of its frequency.
- To estimate the median or quartiles, find the total frequency, use cumulative frequencies to identify which class each measure falls in, then interpolate within that class.
- Always check that the areas of the bars (not just their heights) are being used whenever frequency is being compared or calculated.
Worked example
The frequency table shows the ages, in years, of 90 members of a gym: 0 < a <= 20 (frequency 18), 20 < a <= 30 (frequency 27), 30 < a <= 40 (frequency 30), 40 < a <= 60 (frequency 15). Calculate the frequency density for each class, and state which class has the tallest bar on the histogram.
- Find the class width for each class (upper boundary minus lower boundary): 20, 10, 10, 20.
- Divide each frequency by its class width to find frequency density: frequency density = frequency / class width.
- Calculate: 18/20 = 0.9, 27/10 = 2.7, 30/10 = 3.0, 15/20 = 0.75.
- Compare the four frequency densities: the tallest bar belongs to the class with the highest frequency density.
- Identify the class 30 < a <= 40 as having the highest frequency density (3.0), so it has the tallest bar.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
A frequency table shows the ages, in years, of 70 members of a swimming club: 10 <= a < 20 (frequency 14), 20 <= a < 40 (frequency unknown), 40 <= a < 60 (frequency 16). Find the missing frequency and its frequency density.
A histogram shows the masses, in kg, of parcels handled by a depot in one day, with bars 0 < m <= 5 (frequency density 2), 5 < m <= 15 (frequency density 5), and 15 < m <= 40 (frequency density 1.2). (a) Find the total number of parcels. (b) A parcel is chosen at random. Find the probability that its mass is greater than 15 kg.
The frequency table shows the times, t minutes, spent by 100 customers in a supermarket: 0 < t <= 10 (frequency 34), 10 < t <= 25 (frequency 30), 25 < t <= 40 (frequency 24), 40 < t <= 60 (frequency 12). (a) Show that the median customer time is 18 minutes. (b) Estimate the interquartile range of the customer times.
See real past-paper questions on histograms, organised by topic with official mark schemes →
Free printable worksheet
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