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Histograms

Statistics

Back to frequencies

  • Going the other way, multiply each bar's frequency density by its class width to recover the frequency
  • A common question gives the histogram and an empty table, and asks you to fill the frequencies in
  • Add the frequencies to find the total number of items where the question needs it

Part of a bar

  • Where a question asks about only part of a class, take the matching part of that bar's area
  • The frequency density is constant across the bar, so the frequency for part of it is that frequency density times the part-width
The same histogram with two further blocks shaded green: a bar of density 0.8 from 60 to 70 and a very low wide bar from 80 to 100, showing that it is the area of a block rather than its height that gives the frequency
Source: Drawing histograms by Save My Exams
  • This assumes the data is spread evenly through the class, which is why the answer is an estimate
Exam tip

Frequency is the area of a bar, not its height — the single idea the whole topic rests on. Write the class widths and frequency densities into a table before drawing anything, because the method marks depend on showing you used density rather than frequency. The class width is the difference between the bounds, so 20 < m ⩽ 30 has width 10, not 11. Bars touch, with no gaps.

Worked example

Reading frequencies off a histogram

A histogram of the data above is drawn. Find the frequency for the class 20 < m ⩽ 30, and estimate how many items have a mass between 30 g and 40 g.

A histogram with mass in grams from 0 to 80 along the horizontal axis and frequency density from 0 to 4 up the vertical axis. Four bars of unequal width sit side by side with no gaps: from 0 to 20 at height 1.2, from 20 to 30 at height 3.5, from 30 to 50 at height 1.5, and from 50 to 80 at height 0.7.

Solution:

  • For the whole class, frequency = frequency density × class width
  • The bar for 20 < m ⩽ 30 has frequency density 3.5 and width 10
  • Frequency = 3.5 × 10 = 35 items, matching the original table
  • For 30 g to 40 g, this is part of the class 30 < m ⩽ 50, whose frequency density is 1.5
  • The part-width is 40 − 30 = 10
  • Estimated frequency = 1.5 × 10 = 15 items
  • This is an estimate because it assumes the 30 items in that class are spread evenly across its full width of 20

Comparing two histograms

  • Comparing two histograms side by side is only valid when their intervals match and their vertical scales match
  • Otherwise convert both back to frequencies before saying anything about them