Build a cumulative frequency table from grouped data, plot the curve at the upper bound of each class, and read off the median, the quartiles and how many lie above a value.
A box plot wants five numbers, and last stop you had them because eleven runners meant eleven times you could count along. Wren Lane's open 10k is a different job. Eighty runners finished, and the timing tent handed over classes rather than times: 6 runners in 30 ≤ t < 40, then 14, 26, 22 and 12 across the next four bands of ten minutes. No list to count along. The way in is a running total, and on a graph it has a name: cumulative frequency. The cumulative frequency at any time is how many runners had finished by then.
Add the frequencies as you go down: 6, then 20, then 46, then 68, then 80. Each of those is the number of runners home by the end of its band. Now the plotting rule that matters. Each total is plotted against the upper boundary of its class, not the midpoint. By 40 minutes exactly 6 runners were in; by 50 minutes 20 were. Nothing is known about halfway through a band, so the honest point to mark is the boundary where the count is certain. Turn it round and a single class's own frequency is the jump between two totals. Join the points with a smooth curve.
To read the median, work up the frequency axis, not along the time axis. There are 80 runners, so go across from 40, hit the curve, and drop to the time axis: about 58 minutes. The ordered-list rule would send you to 40.5 rather than 40, a gap thinner than the pencil line, so half the total is the place to look.
The quartiles come off the same axis, with the same rounding as the median. A quarter of 80 is 20, so go across at 20 and drop down: the curve is at exactly 50 minutes there, so the lower quartile is 50. Three quarters of 80 is 60, and reading across at 60 drops to about 66 minutes. The interquartile range is 66 - 50 = 16 minutes. Those plus the outer band boundaries are what a box plot needs.
How many runners took longer than 65 minutes? Go up from 65 to the curve and across: about 57 were home, so 80 - 57 = 23 were still out. The height gives how many were under, so the total minus it gives how many were over. The shape says more. Where the curve climbs steeply, runners are crossing the line thick and fast; where it flattens, hardly anyone is coming in. Readings are estimates, because the curve assumes each band filled steadily.
A cumulative frequency graph plots the running total of the frequencies against the upper boundary of each class, joined with a smooth curve. Read the median by going across at half the total frequency and dropping to the value axis, and the quartiles at a quarter and three quarters. The curve's height at any value is how many are below it, so the total minus that height is how many are above. Every reading is an estimate, because the data arrived in classes.
Wren Lane's open 10k finished an hour ago and the timing tent hands over bands, not times: 6 runners in 30 to 40 minutes, then 14, 26, 22 and 12. Eighty finishers, no list. Mia wants the same five figures she drew a box plot from last week.
First job is the running total. What is the cumulative frequency by 50 minutes?
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.