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9b · E9.6–E9.7

Cumulative frequency and histograms

Topic 9 Statistics · Extended only · Papers 2 and 4

EXTENDED Everything on this page is Extended content only.

🎯What you need to be able to do

  • Complete a cumulative frequency table and draw a cumulative frequency curve.
  • Estimate the median, quartiles, interquartile range and percentiles from the curve, and interpret them.
  • Calculate frequency density, and draw and interpret histograms with equal or unequal class widths.

📚The mathematics

Cumulative frequency

Cumulative frequency is a running total: the number of values up to the end of each class. Plot each running total at the upper boundary of its class (not the midpoint), mark the points with small crosses, start from zero at the lower boundary of the first class, and join them with a smooth curve.

To read the curve with \(n\) values: the median is at \( \tfrac{n}{2} \), the lower quartile at \( \tfrac{n}{4} \), the upper quartile at \( \tfrac{3n}{4} \), and the \(p\)th percentile at \( \tfrac{p}{100} \times n \). Go across from the cumulative frequency axis to the curve, then down. To find how many values are above something, read the curve and subtract from \(n\).

A cumulative frequency curve for the masses of 80 parcels, with crosses at (2, 8), (4, 28), (6, 58), (8, 74) and (12, 80), joined by a smooth S-shaped curve starting from (0, 0). Dashed reading lines at cumulative frequencies 20, 40 and 60 give the lower quartile about 3.3 kg, the median about 4.8 kg and the upper quartile about 6.2 kg.
The worked example: across at 20, 40 and 60, then down, gives LQ \( \approx 3.3 \), median \( \approx 4.8 \), UQ \( \approx 6.2 \) kg.

Histograms

In a histogram the area of each bar, not its height, represents the frequency. The vertical axis is frequency density:

\[ \text{frequency density} = \frac{\text{frequency}}{\text{class width}} \qquad\Longleftrightarrow\qquad \text{frequency} = \text{frequency density} \times \text{class width} \]

Bars touch (the data are continuous) and their widths match the class widths. When the classes are unequal, a tall bar is not necessarily a large frequency: compare areas.

A histogram of the same 80 parcel masses with unequal classes. Frequency densities: 4 for 0 to 2 kg, 10 for 2 to 4 kg, 15 for 4 to 6 kg, 8 for 6 to 8 kg, and 1.5 for the wider class 8 to 12 kg. Each bar is labelled with its frequency.
The last class is twice as wide, so its 6 parcels give a bar only 1.5 high. Areas: \( 2 \times 4 = 8 \), \( 2 \times 10 = 20 \), and so on.

✏️Worked example

The masses \(m\) kg of 80 parcels are: \( 0 < m \le 2 \): 8; \( 2 < m \le 4 \): 20; \( 4 < m \le 6 \): 30; \( 6 < m \le 8 \): 16; \( 8 < m \le 12 \): 6. (a) Complete the cumulative frequency table and draw the curve. [3] (b) Estimate the median and the interquartile range. [3] (c) Estimate how many parcels weigh more than 7 kg. [2] (d) Calculate the frequency densities for a histogram. [2]

(a) Cumulative frequencies at \( m = 2, 4, 6, 8, 12 \): 8, 28, 58, 74, 80. Plot these (and \( (0, 0) \)) and join with a smooth curve.

(b) Median at \( \tfrac{80}{2} = 40 \): about 4.8 kg. LQ at 20: about 3.3 kg; UQ at 60: about 6.2 kg. IQR \( \approx 6.2 - 3.3 = 2.9 \) kg.

(c) At \( m = 7 \) the curve reads about 68, so about \( 80 - 68 = 12 \) parcels weigh more than 7 kg.

(d) \( 8 \div 2 = 4 \), \( 20 \div 2 = 10 \), \( 30 \div 2 = 15 \), \( 16 \div 2 = 8 \), \( 6 \div 4 = 1.5 \).

Check it. The median must lie in the class containing the 40th value: 28 < 40 ≤ 58, the \( 4 < m \le 6 \) class ✓. Readings from a hand-drawn curve vary a little; mark schemes accept a range around each value.
Plotting at midpoints. Cumulative frequency counts everything up to the end of a class, so plot at 2, 4, 6, 8, 12 — not at 1, 3, 5, 7, 10. For the histogram, plotting frequency instead of frequency density makes the wide class look six times too big.

📝Practise

All Extended. Questions 3–5 use the parcel data and curve above.

1. (Non-calculator.) A class \( 10 < t \le 25 \) has frequency 45. Find its frequency density. [1]
\( 45 \div 15 = 3 \).
2. (Non-calculator.) A histogram bar has width 5 and frequency density 2.4. Find the frequency. [1]
\( 2.4 \times 5 = 12 \).
3. (Non-calculator.) Use the curve to estimate the 90th percentile of the parcel masses. [2]
90% of 80 is 72; across at 72 and down gives about 7.5 kg.
4. (Non-calculator.) Estimate how many parcels weigh 3.3 kg or less, and explain the connection with the lower quartile. [2]
About 20 — that is exactly what the lower quartile means: a quarter of the 80 parcels are at or below it.
5. (Calculator.) Calculate an estimate of the mean mass of the 80 parcels. [3]
Midpoints 1, 3, 5, 7, 10: \( \dfrac{8 + 60 + 150 + 112 + 60}{80} = \dfrac{390}{80} = 4.875 \approx 4.88 \) kg.
6. (Non-calculator.) Two classes sat a test. Class A: median 62, IQR 8. Class B: median 58, IQR 21. Make two comparisons. [2]
Class A did better on average (higher median). Class A’s marks were more consistent (smaller IQR); Class B’s were more spread out.

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • Corbettmaths — cumulative frequency and histograms with unequal class widths
  • Khan Academy — interpreting quartiles and interquartile range