Presenting data: histograms, cumulative frequency and box plots

Maths: Applications and Interpretation · Statistics and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A histogram shows continuous data grouped into classes. The bars touch, and when the classes have equal width the height of each bar is the frequency of that class.

A cumulative frequency graph shows a running total: the number of values less than or equal to each upper class boundary. You read it backwards, from the frequency axis across to the curve and down, to estimate the median, the quartiles and percentiles.

A box-and-whisker diagram shows five numbers: minimum, lower quartile, median, upper quartile and maximum. Two box plots on the same scale make it easy to compare the average and the spread of two sets of data.

Rules to remember

Cumulative frequency is plotted against the upper boundary of each class.
For n values on a cumulative frequency graph: median at n/2, Q₁ at n/4, Q₃ at 3n/4.
The pth percentile is read at a cumulative frequency of (p/100) × n.
Number of values greater than x = n − (cumulative frequency at x).
IQR = Q₃ − Q₁ = length of the box; it holds the middle 50% of the data.
Compare distributions with the median (average) and the IQR (spread): a smaller IQR means more consistent values.
Longer right whisker and median nearer Q₁: positive (right) skew. The reverse: negative (left) skew.

Common mistake

Students read the median at half of the highest value on the horizontal axis. Go to n/2 on the cumulative frequency axis, across to the curve, then down to read the value.

Worked example

A cumulative frequency curve for the times of 120 runners passes through (42, 30), (50, 60) and (61, 90), with time in minutes first. Estimate the median, the interquartile range and the number of runners who took more than 61 minutes.

n = 120, so the median is at 120 ÷ 2 = 60: median = 50 minutes.
Q₁ is at 120 ÷ 4 = 30, giving 42; Q₃ is at 3 × 120 ÷ 4 = 90, giving 61.
IQR = 61 − 42 = 19 minutes.
More than 61 minutes: 120 − 90 = 30 runners.

Answer: Median 50 minutes; IQR 19 minutes; 30 runners.

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Topic 4: Statistics and Probability

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