Finding quartiles from a list and from cumulative frequency, the interquartile range, percentiles and identifying outliers.

What this dot point is asking

AQA wants you to find quartiles from an ordered list and from a cumulative frequency curve, calculate the interquartile range, work with percentiles, and use the $1.5 \times \text{IQR}$ rule to decide whether a value is an outlier. This material underpins box plots, so accurate quartiles here feed straight into the diagram questions.

Finding quartiles from a list

For a list of $n$ values that has been sorted, AQA uses these positions: $Q_1$ at $\frac{n+1}{4}$, the median at $\frac{n+1}{2}$, and $Q_3$ at $\frac{3(n+1)}{4}$. When a position is a whole number, read off that value directly. When it falls between two positions, interpolate: a position of $3.5$ means halfway between the $3$rd and $4$th ordered values. Note the small but important difference: for a raw list you use $\frac{n+1}{4}$, but for a cumulative frequency graph of grouped data you use $\frac{n}{4}$, because a continuous curve treats the data as one large block rather than discrete items.

The interquartile range

Because it discards the bottom and top quarters, the interquartile range is the natural partner to the median for skewed data. It also drives the outlier test below and sets the box length on a box plot.

Quartiles from cumulative frequency

For grouped data plotted as a cumulative frequency curve, read $Q_1$ at $\frac{n}{4}$, the median at $\frac{n}{2}$ and $Q_3$ at $\frac{3n}{4}$ on the vertical axis, then go across to the curve and down to the horizontal axis to estimate each value. These are estimates because the original values inside each class are unknown. The points are read where the curve crosses, so a smooth, accurately plotted ogive is essential for marks.

Quartiles also feed directly into other diagrams and statistics. The five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) is exactly what a box plot displays, so accurate quartiles here translate into a correct box plot in the next topic. The interquartile range is the box length, the median position inside the box hints at skew, and the outlier boundaries decide where the whiskers stop. Because the quartiles ignore the most extreme quarter at each end, they give a robust picture of the data that is not thrown off by a single unusual value, which is why they are the standard summary for skewed real-world data such as incomes or waiting times.

Identifying outliers

The standard AQA outlier rule flags any value that sits well outside the central half of the data. The factor $1.5$ is a convention: it sets the boundary far enough from the box that ordinary values are kept but genuinely unusual ones are flagged. A value beyond the boundary is sometimes called a "mild" outlier, and the rule is applied at both ends, so a data set can have low outliers, high outliers, both or none.