Range, quartiles, interquartile range, percentiles, interpercentile and interdecile range; choosing an appropriate measure of spread; pairing a measure of spread with a measure of central tendency.

What this dot point is asking

Edexcel codes 2c.01, 2c.04 and 2c.05 require you to calculate measures of spread: the range, quartiles, the interquartile range (IQR), percentiles, and (Higher tier) the interpercentile and interdecile range. You must choose an appropriate measure for a context and pair a measure of spread with a suitable measure of central tendency. (Standard deviation has its own page.) Spread questions appear constantly because every comparison of two data sets needs one.

The range

The range is quick to find and uses the full extent of the data, but it depends entirely on the two most extreme values, so a single outlier can make it misleading. It is best used as a rough measure or alongside a more robust one.

Quartiles and the interquartile range

The quartiles divide ordered data into four equal parts: $Q_1$ (lower quartile) has a quarter of the data below it, $Q_2$ is the median, and $Q_3$ (upper quartile) has three quarters below it. For a small ordered list of $n$ values, locate $Q_1$ at position $\frac{n+1}{4}$ and $Q_3$ at $\frac{3(n+1)}{4}$.

Because it discards the lowest and highest quarters, the IQR is resistant to outliers, which is why Edexcel pairs it with the median (not the mean). Using the mean with the IQR is explicitly flagged as an inappropriate pairing.

Percentiles, interpercentile and interdecile range

Percentiles split ordered data into $100$ equal parts: the $p$th percentile has $p\%$ of the data below it (so $Q_1$ is the $25$th percentile and the median is the $50$th). Deciles split it into ten parts. Higher tier measures of spread built from these:

• The interpercentile range between two percentiles, for example the $10$th to $90$th interpercentile range $= P_{90} - P_{10}$, the spread of the middle $80\%$.
• The interdecile range $= D_9 - D_1$ (the $90$th minus the $10$th percentile), also the middle $80\%$.

These exclude the most extreme $10\%$ at each end, giving a robust picture of spread while keeping more of the data than the IQR.

Choosing and pairing measures

Match the measure of spread to the measure of central tendency:

• Median with IQR (or interpercentile / interdecile range) for skewed data or data with outliers.
• Mean with standard deviation for roughly symmetric data (see the standard deviation page).

A correct pairing is examined directly, and quoting an average without an appropriate measure of spread loses marks in comparison questions.