Cumulative frequency

• Cumulative frequency is a running total of the frequencies.
• Plot each running total at the upper class boundary, mark it with a small cross, and join the crosses with a smooth curve.
• Read the median at n/2, the lower quartile at n/4 and the upper quartile at 3n/4 on the cumulative frequency axis.
• Interquartile range = upper quartile − lower quartile. A percentile is read the same way, e.g. the 90th percentile at 0.9n.
• Read up from a value to find how many are below it; subtract from the total for how many are above.

A running total

Add each frequency to the total so far. The last cumulative frequency must equal the total number of values, which is a built-in check. Each running total counts every value up to the upper boundary of that class.

Upper boundaries and a smooth curve

Plot cumulative frequency (up the side) against the upper class boundary (along the bottom): here (10, 4), (20, 15), (30, 34), (40, 59), (50, 73), (60, 80). Start the curve at (0, 0), the lower boundary of the first class, where nobody has finished. Mark each point with a small cross and join them with a smooth curve (not ruled lines). The curve always rises or stays level; it has the shape of a stretched S.

Read across, then down

Find the position on the cumulative frequency axis, read across to the curve, then read down to the value. With a large total like 80, use n/2, n/4 and 3n/4 (not n + 1).

How many below, how many above

To find how many values are below a certain value, read up from that value to the curve and across. To find how many are above it, subtract that reading from the total.