Using cumulative frequency graphs

Cumulative frequency (CF for short) means "how many so far". For each value on the horizontal axis, the graph tells you how many data items are at or below that value.

Here is the running data set for this lesson: the masses of 80 parcels at a depot.

Each cumulative frequency is the previous one plus the next frequency: $6$, then $6 + 14 = 20$, then $20 + 24 = 44$, and so on. The last value is the total, $n = 80$.

Each cumulative frequency is plotted at the upper end of its class. "20 parcels weigh 2 kg or less" belongs at mass 2, because only once you reach 2 kg have all 20 been counted.

The graph also starts at $(0, 0)$, the lower end of the first class: no parcel weighs 0 kg or less, so the cumulative frequency there is 0. (Here $1 < m \le 2$ means "more than 1 kg, up to and including 2 kg".)

If you lined all $n$ parcels up from lightest to heaviest:

• the median is the one halfway along, at cumulative frequency $\tfrac{1}{2}n$;
• the lower quartile $Q_1$ is a quarter of the way along, at $\tfrac{1}{4}n$;
• the upper quartile $Q_3$ is three quarters of the way along, at $\tfrac{3}{4}n$.

For $n = 80$ those positions are $40$, $20$ and $60$.

The method: find the position on the cumulative frequency axis, go across to the graph, then down to the horizontal axis and read the value. In an exam, draw those lines on the graph: they are your working.