25.2 Cumulative frequency — Learn

Prerequisites: read a frequency table, plot coordinates, and find medians and quartiles.

Objectives: form cumulative frequencies, draw and read a cumulative frequency graph, and estimate quartiles.

Cumulative frequency is a running total. It gives the number of observations up to and including a boundary.

Example 1. Plot the cumulative frequency graph for the table.

Use the upper class boundaries and cumulative totals: $(10,4),(20,13),(30,18),(40,20)$. Include $(0,0)$, label both axes, and join the points with a smooth increasing curve.

For 20 observations, estimate the median at cumulative frequency $20/2=10$.

Estimate the lower and upper quartiles at cumulative frequencies $20/4=5$ and $3(20)/4=15$.

Read across from each cumulative frequency to the curve, then down to the time axis. The graph gives approximate values because the original data are grouped.

Watch out. Plot the cumulative total at the upper boundary of each class. Plotting ordinary frequencies instead gives a different graph; check that the curve never decreases.

Progress check

The travel-time list has 20 values. Which cumulative frequencies locate its lower quartile, median and upper quartile?

You should now be able to

• calculate running totals from grouped frequencies;
• plot cumulative frequency against upper class boundaries;
• estimate the median and quartiles from a cumulative frequency graph.