Quartiles from the Curve

Forty students on one curve

Forty students each solve the same puzzle, and the time each one takes is recorded in minutes. 5 students take less than 10 minutes, 20 take less than 20 minutes, 35 take less than 30 minutes, and all 40 take less than 40 minutes. These running totals are cumulative frequencies.

The cumulative frequency curve plots each running total at the end of its class, (10, 5), (20, 20), (30, 35) and (40, 40), and starts from (0, 0), because no student took less than 0 minutes. It climbs from 0 to 40 on the cumulative frequency axis, since 40 is the number of students. Every height on that axis is a number of students, and every point on the curve says how many students took less than the time below it.

The curve rises from 0 to 40, the number of students. It is steepest from 10 to 30 minutes, where 30 of the 40 students finished.

The lower quartile: a quarter of the way up

The lower quartile, $Q_1$Q₁, is the value that a quarter of the data lies below. For a list of values it is the median of the lower half. When the data is grouped, the values are gone, but the curve still says how many students took less than any time, so the lower quartile can be read from it.

A quarter of 40 is 10. Go up the cumulative frequency axis to 10, across to the curve, and straight down to the time axis. The reading is about 13 minutes: a quarter of the students, 10 of them, took less than about 13 minutes.

The reading can be checked with arithmetic. The height 10 lies between the plotted points (10, 5) and (20, 20). It is 5 of the 15 steps from 5 up to 20, a third of the way, so the line between those points reaches it a third of the way from 10 to 20 minutes, at 13⅓ minutes, about 13 minutes.

Across at 10, a quarter of 40, then down to about 13 minutes: the lower quartile.

The median and the upper quartile

Halfway up is 20, half of 40. Reading across at 20 lands on the plotted point (20, 20), so the median is 20 minutes: half of the students took less than 20 minutes.

Three quarters of 40 is 30. Reading across at 30 gives the upper quartile, $Q_3$Q₃, the value that three quarters of the data lies below. The height 30 is 10 of the 15 steps from 20 up to 35, two thirds of the way, so the reading is two thirds of the way from 20 to 30 minutes, at 26⅔ minutes, about 27 minutes.

The same three readings work for any curve. With n values, read across at $\frac{n}{4}$n/4 for the lower quartile, $\frac{n}{2}$n/2 for the median and $\frac{3n}{4}$3n/4 for the upper quartile: for 80 students, at 20, 40 and 60. As with the median, a curve is read at $\frac{n}{4}$n/4 and $\frac{3n}{4}$3n/4 rather than at the positions used for a list, because a reading from a curve is an estimate and for a large group the difference is very small.

Across at 20, half of 40, then down to 20 minutes: the median.

Across at 30, three quarters of 40, then down to about 27 minutes: the upper quartile.

The interquartile range from the curve

The interquartile range is the upper quartile minus the lower quartile: IQR $=Q_3-Q_1=26\frac{2}{3}-13\frac{1}{3}=13\frac{1}{3}$= Q₃ − Q₁ = 26⅔ − 13⅓ = 13⅓ minutes, about 13 minutes. The middle half of the students, the 20 between the 10th and the 30th, took between about 13 and about 27 minutes.

The four quarters of the students do not cover equal stretches of time. The first quarter spans 0 to 13⅓ minutes and the last spans 26⅔ to 40 minutes, 13⅓ minutes each, while the two middle quarters span only 6⅔ minutes each. Where the curve is steep, many students are packed into a short stretch of time, and the quartiles close in on the median. A curve that is steep through the middle gives a small interquartile range, and one that climbs gently gives a large one.

The same 40 students. The reading starts at 30, three quarters of the way up, on the upper quartile, 26.7 minutes. Drag it down to 10, a quarter of the way up, for the lower quartile, 13.3 minutes. Under the axis, the figure prints how many minutes the next 10 students above the reading span. From the lower quartile it is 6.7 minutes, half of the 13.3 minutes that the first 10 students span.

Why a reading from the curve is an estimate

The curve is exact only at the plotted points. The table says that exactly 20 students took less than 20 minutes, but it does not say how the 15 students in the class from 10 to 20 minutes are spread inside it. Joining the points with a straight line assumes that those 15 students are spread evenly across the class, and a smooth curve through the same points assumes something slightly different.

So a quartile read from the curve is an estimate. Two people who draw the curve by hand may read the lower quartile as 13 minutes and as 14 minutes, and both are reasonable. If the 40 times had been kept as a list, the quartiles could be found exactly; once they are grouped, the curve gives the best estimate the table allows. That is why a quartile from a curve is given as “about 13 minutes”.