Draw and read histograms where the classes have different widths, using frequency density so that area, not height, shows the frequency.
Last stop the curve gave five figures from 80 finishers. Wren Lane's next 10k has 80 again, and this time the bands are not all the same width. The tent split the fast end finely and the slow end coarsely: 8 runners in 30 ≤ t < 40, then 12 in 40 ≤ t < 45, 15 in 45 ≤ t < 50, 25 in 50 ≤ t < 60 and 20 in 60 ≤ t < 80. Widths: 10, 5, 5, 10 and 20 minutes. Draw those as plain bars and the picture lies: a count over 20 minutes is not comparable with one over 5.
Look at the widest band and one of the narrowest. The 60 to 80 band holds 20 runners spread across 20 minutes, so about one runner a minute. The 40 to 45 band holds 12 runners across 5 minutes, so nearly two and a half a minute. The second band is more than twice as busy, and the plain chart draws it three fifths the height of the first. Height alone cannot be trusted once the widths differ. The repair is to let the area of each bar carry the frequency instead of the height.
Area of a rectangle is width times height, and the width is already fixed by the class. So the height has to be whatever makes that multiplication come out as the frequency: frequency divided by class width. That height is called the frequency density, and it is how many values fall in one unit of the scale. For the five bands: 8 / 10 = 0.8, 12 / 5 = 2.4, 15 / 5 = 3.0, 25 / 10 = 2.5 and 20 / 20 = 1.0. Read it back the other way and density times width returns the frequency every time.
Because area is the frequency, you can ask about part of a band. How many finished between 50 and 55? That is inside the 50 to 60 band, whose density is 2.5, and the piece is 5 minutes wide: 2.5 x 5 = 12.5, so about 13 runners. A question can cross a boundary too. Between 42 and 48 you take 3 minutes of the 2.4 band and 3 minutes of the 3.0 band: 2.4 x 3 = 7.2 and 3.0 x 3 = 9, giving about 16 runners in total.
Two things to keep hold of. Every part-of-a-band answer assumes the runners were spread evenly inside that band, the same assumption the curve made last stop, so it is an estimate. And the areas of all five bars add up to 80, because between them they account for every runner. The bars sit side by side with no gaps, because the bands run straight into one another along a continuous scale.
When classes have different widths, the height of a bar stops being trustworthy and the area takes over the job. Frequency density is frequency divided by class width, and it is what goes up the vertical axis. Density times width gives the frequency back, so the area of any piece of a bar estimates how many values fall in that stretch. The total area is the total frequency, and every part-of-a-class answer assumes an even spread inside the class.
Last week the curve turned five bands into five figures. A new Wren Lane 10k, 80 finishers, and this time the bands are uneven. The timing tent split the fast end finely and the slow end coarsely: 8 runners in 30 to 40 minutes, 12 in 40 to 45, 15 in 45 to 50, 25 in 50 to 60 and 20 in 60 to 80. Sam draws the five frequencies as five plain bars and stands back, pleased.
Compare the 60 to 80 band with the 40 to 45 band. Which was busier, minute by minute?
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