Cumulative frequency diagrams (discrete and grouped); estimating the median, quartiles and percentiles; box plots; comparing distributions using box plots and the interquartile range.

What this dot point is asking

Edexcel codes 2a.03, 2c.01 and 2c.05 require you to draw and read cumulative frequency diagrams (discrete and grouped), estimate the median, quartiles and percentiles from them, draw and interpret box plots, and compare distributions using a median and a measure of spread. Cumulative frequency curves and box plots are among the most reliable sources of marks in the qualification, because the method is the same every time.

Cumulative frequency diagrams

The crucial plotting rule is to use the upper class boundary, because by the top of a class all of its values have been counted. The curve always rises and levels off at the total $n$. Plotting against midpoints is a common and costly error.

Estimating median, quartiles and percentiles

From a cumulative frequency curve you read values horizontally across, then down:

• Median: go up to the $\frac{n}{2}$th value, across to the curve, down to the axis.
• Lower quartile ($Q_1$): use the $\frac{n}{4}$th value.
• Upper quartile ($Q_3$): use the $\frac{3n}{4}$th value.
• A percentile: the $p$th percentile uses the $\frac{p}{100} \times n$th value (the $90$th percentile uses $0.9n$).

Edexcel uses $\frac{n}{2}$, $\frac{n}{4}$ and $\frac{3n}{4}$ on the cumulative frequency axis (not $\frac{n+1}{2}$, which is for a small ordered list). Because the curve gives an estimate, slightly different sensible readings are accepted.

Box plots

A box plot (box and whisker diagram) summarises a distribution with five numbers: minimum, lower quartile, median, upper quartile and maximum. The box spans $Q_1$ to $Q_3$ (the middle $50\%$ of the data), a line inside marks the median, and the whiskers reach out to the minimum and maximum. The width of the box is the interquartile range $IQR = Q_3 - Q_1$, a measure of spread that ignores extreme values.

Comparing distributions

When two box plots (or two cumulative frequency curves) are compared, Edexcel expects two comparisons in context: one of an average (compare the medians) and one of spread (compare the IQRs or ranges). A higher median means higher values on average; a smaller IQR means more consistent values. Always finish with a sentence relating the comparison to the situation.

A box plot also reveals skewness at a glance. If the median sits closer to the lower quartile and the upper whisker is longer, the data has positive skew (a longer tail of high values); if the median is closer to the upper quartile with a longer lower whisker, the data has negative skew. A median in the centre of a symmetric box suggests a roughly symmetric distribution. Edexcel may ask you to comment on the shape as well as the average and spread, so look at where the median falls within the box.

Using a cumulative frequency graph to estimate proportions

Beyond the median and quartiles, a cumulative frequency curve lets you estimate how many values lie above or below a given figure. To find the number of values below a value $v$, go up from $v$ on the horizontal axis to the curve and across to read the cumulative frequency; to find the number above $v$, subtract that reading from the total $n$. For example, with $n = 200$ apples, if $150$ have a mass of $140$ g or less, then $200 - 150 = 50$ apples are heavier than $140$ g. Converting such a count to a percentage or a probability is a frequent follow-up.