Cumulative frequency tables and graphs, estimating the median and quartiles, and drawing and interpreting box plots.

What this dot point is asking

AQA wants you to build a cumulative frequency table and graph, use it to estimate the median, quartiles and percentiles, calculate the interquartile range, and draw, read and compare box plots, including identifying outliers. Cumulative frequency and box plots are the standard tools for comparing two grouped data sets, so they appear on almost every paper.

Cumulative frequency tables and graphs

The reason you plot against the upper boundary is that the running total counts everything up to and including that boundary, so the point belongs at the top of the class, not the middle. The resulting curve is S-shaped (an ogive) and rises to the total $n$ at the end. It lets you estimate, for any value, how many data items fall below it, which is the basis for reading off the median and quartiles.

Estimating the median and quartiles

Note that for a graph you use $\frac{n}{2}$, $\frac{n}{4}$ and $\frac{3n}{4}$, not the $\frac{n+1}{2}$ positions used for a short raw list. This is because a cumulative frequency curve treats the data as one continuous block of $n$ items rather than discrete points. To read a percentile, use the same idea: the $p$th percentile is read at cumulative frequency $\frac{p}{100} \times n$.

Box plots and outliers

Box plots are ideal for comparing two distributions on one scale: compare medians for the typical value and the interquartile range (the box length) or range (the whisker span) for the spread, always in context. The position of the median line inside the box also signals skew: a median nearer the lower quartile suggests positive skew.

The two diagrams work together: the cumulative frequency curve is where you read the quartiles from grouped data, and the box plot is how you display and compare them. To draw a box plot you need the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum), and the first three quartiles come straight off the curve at $\frac{n}{4}$, $\frac{n}{2}$ and $\frac{3n}{4}$. A cumulative frequency curve can also answer "how many scored more than $40$?" type questions: read up from $40$ to the curve and across to find how many fall below, then subtract from $n$ for the number above. This makes the curve the workhorse for any "estimate the number/percentage above or below a value" question on grouped data.