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Cambridge IGCSE Mathematics — 0580 Extended

Topic 9.3: Statistics — Averages and Spread

For a raw list, Extended 0580 wants the mean, median, mode, quartiles, range and interquartile range, and a reason for which average you used. Order the list first. The mean uses every value; the median is a position; the mode is a frequency.

Mean, median, mode, range

Mean \(= \Sigma x / n\). Median: middle value, or the mean of the two middle values when \(n\) is even. Mode: the value that appears most. Range \(=\) largest \(-\) smallest.

Method

  1. Order the list. Count \(n\).
  2. Mean: add, then divide by \(n\).
  3. Median: position \((n+1)/2\). If that is halfway between two values, average them.
  4. Mode and range from the ordered list.

For 2, 3, 5, 5, 6, 8, 9, 10, 12, 20 find the mean, median, mode and range.

Ten ordered values with mean 8, median 7, mode 5 and range 18
\(\Sigma x = 80\), so the mean is 8. The 5th and 6th values average to 7.
Number line of the ten values with the mean at 8 as a balance point
The 20 on the right pulls the mean above the median. The median barely moves.

Quartiles and IQR

Split the ordered list in half. \(Q_1\) is the median of the lower half; \(Q_3\) is the median of the upper half. When \(n\) is odd, drop the overall median from both halves. \(\mathrm{IQR} = Q_3 - Q_1\).

Ten values split into halves giving Q1 = 5, Q3 = 10 and IQR = 5
Lower half 2, 3, 5, 5, 6 → \(Q_1 = 5\). Upper half 8, 9, 10, 12, 20 → \(Q_3 = 10\).

Which average?

Use the mean when there is no extreme value. Use the median when one value sits far from the rest — that 20 is the reason. Use the mode for qualitative data, or when a value clearly repeats.

When to use mean, median, mode and IQR
A question that says ‘typical’ after showing an outlier wants the median.

A full worked list

3, 5, 5, 6, 8, 9, 13. Find the mean, median, mode, range and IQR.

Seven values with mean 7, median 6, mode 5, range 10 and IQR 4
\(n\) is odd, so drop 6 from each half: \(Q_1 = 5\), \(Q_3 = 9\), IQR = 4.

Paper 2 (non-calculator)

\(49 \div 7 = 7\) exactly. \(80 \div 10 = 8\). Leave integers as integers.

Try this

From 2, 3, 5, 5, 6, 8, 9, 10, 12, 20 find the IQR.

Show answer
Answer

  1. \(Q_1 = 5\), \(Q_3 = 10\).

    \[ 10 - 5 = 5 \]

Exam Traps

  • The mean of an even list’s two middle values is the median — do not then also include those two values as a third “middle”.

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