A-Level Mathematics May/June 2024 Q3(a): Draw a histogram to represent this information.

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  To draw a histogram, we must first calculate the frequency density for each class. The formula is:

  $\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$

  We can construct a table to find the frequency densities:

  Height (h cm)	Frequency	Class Width	Frequency Density Calculation	Frequency Density
  130 ≤ h < 150	16	20	$16 \div 20$	0.8
  150 ≤ h < 160	32	10	$32 \div 10$	3.2
  160 ≤ h < 170	76	10	$76 \div 10$	7.6
  170 ≤ h < 175	64	5	$64 \div 5$	12.8
  175 ≤ h < 195	12	20	$12 \div 20$	0.6

  Now, we draw the histogram based on these values.

  • The horizontal axis is labelled 'Height (h cm)' and has a linear scale from 130 to 195. Key values such as 130, 150, 170, 190 are marked.
  • The vertical axis is labelled 'Frequency Density' and has a linear scale from 0 to 14. Key values such as 0, 5, 10 are marked.

  The bars are drawn with the following widths and heights:

  1. A bar from 130 to 150 with a height of 0.8.
  2. A bar from 150 to 160 with a height of 3.2.
  3. A bar from 160 to 170 with a height of 7.6.
  4. A bar from 170 to 175 with a height of 12.8.
  5. A bar from 175 to 195 with a height of 0.6.

  (A correctly drawn graph with the features described above would be presented here.)

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  How the marks are awarded

  • M1 — Calculating at least four out of the five correct frequency densities (0.8, 3.2, 7.6, 12.8, 0.6), as shown in the working table.
  • A1 — Drawing all five bars with the correct heights corresponding to the calculated frequency densities, on a correctly scaled and labelled vertical axis.
  • B1 — Drawing all five bars with the correct widths and positions (130-150, 150-160, 160-170, 170-175, 175-195) on a correctly scaled and labelled horizontal axis.
  • B1 — Correctly labelling both axes, for example as 'Frequency Density' and 'Height (h cm)'.

  Common mistakes

  • Plotting frequency on the vertical axis instead of frequency density. This is the most common error and shows a misunderstanding of the principle of histograms.
  • Assuming all bars have the same width and not scaling the horizontal axis correctly according to the given class intervals.
  • Calculating class widths incorrectly, for example using a width of 19 for the 130-150 class instead of 20, leading to incorrect frequency densities.
  • Forgetting to label one or both axes, or using incorrect labels such as 'Frequency' for the vertical axis.

  Examiner tip: For any histogram with unequal class widths, always remember that the vertical axis must be frequency density, calculated as frequency divided by class width.

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