Cumulative Frequency

How many are below a value?

Nineteen students record how long they spent on a homework task, and the times are grouped into classes. Writing t for a time in minutes, 3 students are in the class $0\le t<10$0 ≤ t < 10, 8 are in $10\le t<20$10 ≤ t < 20, 6 are in $20\le t<30$20 ≤ t < 30 and 2 are in $30\le t<40$30 ≤ t < 40.

Each frequency says how many students are in one class, and on its own the table answers questions about one class at a time. Many questions are about everything below a value instead. How many students took less than 20 minutes? That is the first two classes together: 3 + 8 = 11 students.

On their own the classes hold 3, then 8, then 6, then 2 students.

A running total

Keep a running total down the frequency column: 3, then 3 + 8 = 11, then 11 + 6 = 17, then 17 + 2 = 19. Each running total is a cumulative frequency, the number of values up to the end of that class.

So 3 students took less than 10 minutes, 11 took less than 20 minutes, 17 took less than 30 minutes, and all 19 took less than 40 minutes. The last cumulative frequency is always the total number of values, which is a check on the adding.

A cumulative frequency never goes down. Each step adds the next frequency, and a frequency is never negative, so the running total can only rise or stay level.

The running total, or cumulative frequency, grows 3, 11, 17, 19 and ends at the 19 students.

Plot each total at the end of its class

To draw a cumulative frequency curve, plot each running total above the upper boundary of its class, the value where the class ends: (10, 3), (20, 11), (30, 17) and (40, 19). Start from (0, 0), because no student took less than 0 minutes. Then join the points with a smooth curve or with straight lines.

The upper boundary is the only correct place. The total 11 counts every student who took less than 20 minutes, and that is only true once the whole class $10\le t<20$10 ≤ t < 20 has been counted, at 20. Plotted at the midpoint, 15, the point would claim that 11 students took less than 15 minutes, which the table does not say: some of the 8 students in that class took longer than 15 minutes. Plotting at midpoints shifts the whole curve half a class to the left.

The running totals 3, 11, 17 and 19 plotted at the class ends 10, 20, 30 and 40 minutes. The curve climbs to 19, just below the top of the axis at 20.

The shape of the curve

The curve never falls. Where a class holds many values the curve is steep: from 10 to 20 minutes it climbs 8 students. Where a class holds few values it is nearly flat: from 30 to 40 minutes it climbs only 2.

Many data sets have few values at each end and most in the middle, so their curves start flat, climb steeply through the middle and level off at the top, in a stretched S shape. The homework curve has that shape: it climbs 3, then 8, then 6, then 2. The top of the curve is always level with the total number of values.

Reading how many are below a value

The curve answers the question “how many are below this value?” for any value, not only at the class ends. How many students took less than 25 minutes? Go up from 25 on the time axis to the curve, then across to the cumulative frequency axis. The reading is 14.

The arithmetic behind the reading: 25 minutes is halfway through the class $20\le t<30$20 ≤ t < 30, which holds 6 students, so the curve is halfway from 11 to 17 there, at 11 + 3 = 14. It is an estimate, because it assumes the 6 students are spread evenly across the class. So about 14 students took less than 25 minutes, and about 19 − 14 = 5 took 25 minutes or more.

The dashed lines join 25 minutes on the time axis to 14 on the cumulative frequency axis: about 14 students took less than 25 minutes.

Looking ahead: where the median sits

The curve can also be read the other way, from a number of students to a time. The median is the time of the middle student. Half of 19 is 9.5, so go in at 9.5 on the cumulative frequency axis, across to the curve, then down to the time axis. The reading, about 18 minutes, is the median time. On a curve the median is read at half the total, $\frac{n}{2}$n/2, rather than at position $\frac{\left(n+1\right)}{2}$(n + 1)/2 as in a list: the curve gives an estimate, and for a large group the two readings are almost the same.

The same move at a quarter of the total and at three quarters of it gives the lower and the upper quartile, and their difference is the interquartile range. For a question about the top of the data, such as the top 10% of a group of 80 students, count from the bottom first: 10% of 80 is 8 students, and 80 − 8 = 72 students are below them, so go in at 72.

Across at 9.5, half of the 19 students, and down to about 18 minutes: the median time.

A curve for 40 people. Drag the reading up and down the side: at 20, half of 40, the reading lands on the median, 20 minutes. Bring it down to 10, a quarter of 40, for the lower quartile.