Five-number summary

The five-number summary is five values quoted in ascending order: the minimum, the first quartile $Q_1$, the median, the third quartile $Q_3$, and the maximum. All five carry the units of the data, and together they cut the ordered values into four groups of about a quarter each. This site finds the quartiles the way a TI-84 does, taking the median of the lower half and the median of the upper half with the overall median left out of both when the count is odd.

For the nine values 4, 7, 8, 11, 12, 15, 16, 20, 31 the median is the fifth, 12. The lower half is 4, 7, 8, 11, so $Q_1 = (7 + 8)/2 = 7.5$. The upper half is 15, 16, 20, 31, so $Q_3 = (16 + 20)/2 = 18$. The summary is 4, 7.5, 12, 18, 31, and the interquartile range is $18 - 7.5 = 10.5$.

Now the reading error, the one boxplots invite. The stretch from $Q_3 = 18$ to the maximum of 31 is 13 units wide, while $Q_1 = 7.5$ to the median of 12 is 4.5 units wide, so the first looks like it holds far more data. It does not. Both hold about a quarter of the values. Width shows spread, not count: a wide section means those values are strung out, a narrow one means they are packed tight.

The five numbers also cannot report shape. They carry nothing about peaks, so a set with two clear clumps and one with a single central pile can produce the same summary. Claims about modes must come from a dotplot, histogram, or stemplot.

One warning if you check work against software. The median-excluded rule is a convention, not the only one. A spreadsheet's inclusive quartile function interpolates instead, ranking at $k(n-1) + 1$, and on these same nine values it returns $Q_1 = 8$ and $Q_3 = 16$, an IQR of 8 rather than 10.5. This site and the TI-84 use the rule above. Summary statistics and their displays are Unit 1 topics 1.7 and 1.8.