Cumulative frequency: curves, polygons and linear interpolation

Last time

Before we start: last time, in Frequency tables, class boundaries, histograms and frequency polygons, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Running totals

Cumulative frequency is a running total. Plotted on a graph, it gives the median, the quartiles and percentiles of grouped data, and how many values lie above or below any point. This lesson covers drawing the curve or polygon, reading it, and linear interpolation.

Cumulative frequency tables

Add the frequencies as you go: the cumulative frequency at a boundary is the number of values below it. The waiting times of 100 customers: 20 waited under 10 minutes, 40 from 10 to 20, 25 from 20 to 30, and 15 from 30 to 40. The cumulative frequencies at 10, 20, 30 and 40 minutes are 20, 60, 85 and 100.

The table, worked

The waiting times with their running totals.

The cumulative frequency polygon

Time along the bottom, cumulative frequency up the side. Each total is plotted at the upper class boundary, starting from zero at 0 minutes, and the points are joined with straight lines. A smooth curve through the same points is a cumulative frequency curve.

Reading the graph

For n values, read the median at a cumulative frequency of n over 2, the lower quartile at n over 4, and the upper quartile at 3n over 4. A percentile works the same way: the 90th percentile is read at 90 per cent of n. To find how many lie above a value, read the cumulative frequency there and subtract it from n.

Linear interpolation

Assuming the values are spread evenly within a class, the median is L plus n over 2 minus F, over f, times w. L is the lower boundary of the median class, F the cumulative frequency before it, f its frequency, and w its width. Here the 50th value is in the class 10 to 20: 10 plus 30 over 40 times 10 is 17.5 minutes.

Quartiles by interpolation

The lower quartile is the 25th value, in the class 10 to 20: 10 plus 5 over 40 times 10, which is 11.25 minutes. The upper quartile is the 75th value, in the class 20 to 30: 20 plus 15 over 25 times 10, which is 26 minutes. So the interquartile range is 26 minus 11.25, which is 14.75 minutes.

Curve or polygon?

Join the points with straight lines for a polygon, or with a smooth curve, as the question asks. Linear interpolation gives the same answers as reading a polygon, because both assume the values are spread evenly. Readings from a graph are estimates, as the exact values in each class are unknown.

Mistakes that lose marks

Mistake one: plotting cumulative frequencies at the mid-points instead of the upper boundaries. Mistake two: forgetting the starting point, a cumulative frequency of zero at the lowest boundary. Mistake three: in interpolation, using the cumulative frequency of the median class itself as F. The fix: upper boundaries, a starting point, and F from the class before.

The full-marks tip

The full-marks tip: show your reading lines on the graph, across from the cumulative frequency and down to the axis. In interpolation, write the formula with the numbers in place before working it out.

On one card

So, on one card: cumulative frequency. Keep it for revision.