Calculating the five-figure summary (minimum, lower quartile, median, upper quartile, maximum), the range and interquartile range, drawing boxplots, and comparing two data sets.

What this dot point is asking

The SQA wants you to summarise a data set with the five-figure summary, calculate the median, quartiles, range and interquartile range, draw and read a boxplot, and compare two data sets by referring to both an average and a measure of spread.

The median and quartiles

Put the data in order from smallest to largest. The median is the middle value; if there are two middle values, take their mean. The quartiles split the data into four equal parts: the lower quartile $Q_1$ is the median of the lower half, and the upper quartile $Q_3$ is the median of the upper half.

The range and interquartile range

The range (maximum minus minimum) measures the full spread but is sensitive to extreme values. The interquartile range measures the spread of the middle half and so resists outliers.

Boxplots

A boxplot is a picture of the five-figure summary. A box stretches from $Q_1$ to $Q_3$ with the median drawn as a line inside it; whiskers extend from the box to the minimum and maximum. The width of the box is the IQR, and the position of the median line shows whether the data leans high or low.

Comparing two data sets

A good comparison always mentions both an average and a spread, and puts the conclusion in context. Compare the medians to say which set is typically higher, then compare the IQRs to say which set is more consistent.

The phrasing matters in the exam: name the measure, give the values, and finish with a conclusion in the context of the question, such as "more consistent" or "typically higher", rather than just quoting numbers.

Examples in context

These tools compare groups fairly. A teacher comparing two classes' marks looks at the median to see which class did better typically and the IQR to see which was more consistent. Sports analysts compare players' performances over a season using boxplots, where a narrow box shows a reliable performer. Quality control uses the IQR to monitor whether a process stays consistent.

Try this

Q1. Find the median of $2, 5, 7, 9, 11$. [1 mark]

• Cue. The middle value is $7$.

Q2. Find the IQR of the summary min $4$, $Q_1 = 6$, median $9$, $Q_3 = 13$, max $18$. [1 mark]

• Cue. $13 - 6 = 7$.

Q3. Set X has median $40$, IQR $5$; Set Y has median $40$, IQR $12$. Compare them. [2 marks]

• Cue. Same average, but X is more consistent (smaller IQR).