Range, interquartile range, percentiles, the effect of outliers, and choosing a measure of spread.

What this dot point is asking

AQA wants you to measure the spread of data using the range, interquartile range and percentiles, understand how outliers affect each measure, and choose a suitable measure of spread for a situation. Spread questions almost always pair with an average: examiners reward candidates who quote one average and one spread, both in context.

The range

The range is the crudest measure of spread because a single unusually large or small value sets it entirely. It is fine for small, clean data sets, or for stating the full extent of variation (for example the range of daily temperatures), but it tells you nothing about how the bulk of the data is distributed.

The interquartile range

Because it cuts off the bottom and top quarters, the interquartile range is resistant to outliers, which makes it the standard companion to the median for skewed data. A small interquartile range means the central half of the data is tightly grouped; a large one means even the middle values are widely spread. On AQA box plot and cumulative frequency questions, the interquartile range is read directly from $Q_1$ and $Q_3$.

Percentiles

To find the position of the $p$th percentile in a list of $n$ ordered values, AQA uses $\frac{p}{100} \times n$. For $n = 80$ the $90$th percentile is at position $0.9 \times 80 = 72$, so you read off the $72$nd value (or interpolate from a cumulative frequency graph for grouped data). Interpercentile ranges are popular in exams because they let you describe spread while deliberately discarding the volatile tails.

The effect of outliers and choosing a measure

The choice rule examiners reward: use the range for a quick measure of total spread on clean data, the interquartile range when there are outliers or the data is skewed (pair it with the median), and the standard deviation when the data is roughly symmetrical and you want a measure that uses every value (pair it with the mean).

The reason this matters is that a measure of spread is only useful if it reflects the typical variation rather than one freak value. The range uses only the two most extreme values, so it is the least robust. The interquartile range and interpercentile ranges deliberately discard the tails, so they are robust but ignore some information. The standard deviation uses every value, so it is the most informative, but it pays for that by being sensitive to outliers. Choosing the right one, and pairing it with the matching average, is exactly the judgement examiners test in "compare the data" questions, where you must quote one average and one spread and justify the choice in context.