Cumulative frequency diagrams

Maths · Statistics. 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

Cumulative frequency is a running total: it tells you how many values are less than or equal to a given value. To make the table, add each frequency to the total of all the frequencies before it. The last cumulative frequency equals the total number of values.

On the diagram, each cumulative frequency is plotted at the upper end of its class, and the points are joined to make a curve that never goes down.

The curve lets you estimate the median and the quartiles. Find the position on the cumulative frequency axis, read across to the curve, then down to the data axis. The interquartile range measures the spread of the middle half of the data.

Rules to remember

Cumulative frequency = sum of all the frequencies up to the end of that class.
Plot each point at (upper class boundary, cumulative frequency).
For n values: median at n/2, lower quartile at n/4, upper quartile at 3n/4 on the cumulative frequency axis.
Interquartile range = upper quartile − lower quartile.
The pth percentile is read at (p ÷ 100) × n. The median is the 50th percentile.
Number of values greater than x = total − cumulative frequency at x.
Frequency of one class = its cumulative frequency − the cumulative frequency of the class before.

Common mistake

Giving the position as the answer, for example writing 40 as the median of 80 values. 40 is only where you start on the cumulative frequency axis; read across to the curve and down to the data axis to get the median.

Worked example

A cumulative frequency curve shows the masses of 200 apples. Reading across from cumulative frequencies 50, 100 and 150 gives masses of 92 g, 104 g and 118 g. The curve also shows that 170 apples have a mass of 125 g or less. Find the interquartile range and the number of apples heavier than 125 g.

Quartile positions: 200 ÷ 4 = 50 and 3 × 50 = 150, so lower quartile = 92 g and upper quartile = 118 g.
Interquartile range = 118 − 92 = 26 g. (104 g is the median.)
Heavier than 125 g: 200 − 170 = 30 apples.

Answer: Interquartile range = 26 g; 30 apples are heavier than 125 g

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