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Cambridge IGCSE Mathematics — 0580 Extended

Topic 9.7: Statistics — Cumulative Frequency

A cumulative frequency is a running total. On 0580 you plot it at the upper class boundary as a small cross, then join with a smooth increasing curve. From the graph you estimate the median, percentiles, quartiles and IQR.

The running-total table

Add frequencies down the table. The last cumulative frequency must equal \(n\). Plot \((\text{upper bound}, \text{CF})\), and include the lowest bound at CF \(= 0\).

Method

  1. Write the upper bound of each class.
  2. CF column: add as you go.
  3. Plot the points as \(\times\). Join with a smooth curve — not a rulered polyline.
Cumulative frequency table of 80 heights with upper bounds and running totals
Points: (150, 6), (160, 20), (170, 48), (180, 70), (190, 80), and (140, 0).

The curve

Cumulative frequency curve of 80 heights plotted at upper class boundaries
The curve starts at the lowest bound with CF = 0 and finishes at (190, 80).

Median, quartiles, IQR

From a CF graph of grouped data, use \(n/2\), \(n/4\) and \(3n/4\) — not \((n+1)/2\). That \((n+1)/2\) rule was for a raw list.

\(n = 80\). Estimate the median height.

Construction lines from cumulative frequency 40 down to about 167 cm
Read across from 40, then down. Median ≈ 167 cm.
Q1 at CF 20 and Q3 at CF 60 read as 160 cm and 175 cm
IQR ≈ 175 − 160 = 15 cm. The middle 50% sit in a 15 cm band.

Percentiles

The \(k\)th percentile is the value with \(\mathrm{CF} = (k/100)\times n\). For 80 students the 90th percentile is read at CF \(= 72\).

90th percentile construction from CF 72 down to about 182 cm
90% of the students are shorter than about 182 cm.

Paper 2 (non-calculator)

\(n/2 = 40\), \(n/4 = 20\), \(3n/4 = 60\). Those CF readings are exact integers here; the heights you read off the scale are estimates.

Try this

For these 80 heights, at what cumulative frequency do you read \(Q_3\)?

Show answer
Answer

  1. \(3n/4 = 3\times 80 / 4\).

    \[ 60 \]

Exam Traps

  • Plotting CF at the class midpoint (or at the lower bound) shifts the whole curve. The median then comes out wrong even if the reading technique is perfect.

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