Cumulative Frequency Curves

You compared numerical summaries using medians and quartiles. Now use grouped race times. Wren Lane's 80 finishers have frequencies 6, 14, 26, 22 and 12 in the five classes below. Cumulative frequency is a running total. For these right-closed classes, the cumulative height at a boundary counts times at or below it. A point (time, count) puts time horizontally and count vertically.

Keep adding the frequencies: 6, 20, 46, 68, 80. The final height is the total number of observations. Recover a class frequency by subtracting neighbouring totals: 46 - 20 = 26 in the class ending at 60. Frequencies are zero or positive, so cumulative totals cannot decrease. An empty class leaves the total unchanged.

The final height N gives the group size. For the median estimate, find N/2 on the frequency axis, read across to the graph, then down to the value axis. For 80 runners, use height 40; the time estimate is about 58 minutes. Do not halve the largest time. Different group sizes need different heights: totals 40 and 200 use heights 20 and 100. Their median estimates can still match.

An at-or-below height counts values up to and including x. Subtract it from N to estimate the count above x. At 65 minutes, height 57 gives 80 - 57 = 23 later finishers. Reverse this using height = N - count above. For a < x ≤ b, subtract height at a from height at b. Add disjoint lower and upper groups to count both ends.

On paper, label both axes and use equal spacing for equal amounts. Plot each cumulative total at its class's upper boundary, not its midpoint. Include zero at the first lower boundary. Join points without falling, using straight segments or a nondecreasing smooth curve. A table does not supply midpoint counts. For classes a ≤ x < b, the total at b counts values below b; check which endpoints are included.

Straight segments model an even spread within each class. To interpolate, find the fraction of the frequency rise needed, then move that fraction of the class width. The race median height 40 lies between (50, 20) and (60, 46). The fraction is (40 - 20)/(46 - 20) = 20/26. The time estimate is 50 + (20/26) × 10 ≈ 57.7 minutes. A smooth curve may give a slightly different estimate.

The target needs 20 of the class's 26 added runners. Use the fraction 20/26 of the ten-minute class width.

Boundary totals are known from the table; positions within classes are not. Grouped median and quartile readings estimate raw statistics. Different raw lists can fit the same table. The actual minimum lies inside the first nonempty class, and the maximum inside the last. Boundaries need not be actual observations. A box plot with actual extremes needs those values separately; using outer boundaries gives an approximation.

A steep rise means many observations in a short value interval. A steady straight rise models equal numbers in equal-width intervals. An exactly horizontal stretch adds no observations; a nearly horizontal stretch adds few. For an at-or-below graph, the count in 20 < x ≤ 30 is height at 30 minus height at 20. Compare curve shapes on matching scales and account for group sizes.

Cumulative relative frequency uses proportions instead of counts. Divide a cumulative count by N to get a proportion; multiply a proportion by N to recover a count. For example, 0.6 × 80 = 48. Read quartiles at 0.25 and 0.75 and the median at 0.5, then across to the graph and down to the value axis. A percentage scale uses 25%, 75% and 50%.

Three fifths is 0.6. For 80 observations, this share represents 0.6 × 80 = 48 observations.