Five-Number Summary vs Boxplot

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.

A boxplot draws exactly five numbers: the minimum, $Q_1$ (Q one, the first quartile), the median, $Q_3$, and the maximum. The box runs from $Q_1$ to $Q_3$, so its length is the interquartile range. In the modified version, the one normally drawn, any value beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$ is plotted as its own point and the whiskers stop at the most extreme values inside those fences. Quartiles here follow the median-excluded (TI-84) rule.

Two sets of eight values show what those five numbers cost. Set A is 10, 20, 30, 40, 60, 70, 80, 90. Set B is 10, 24, 26, 27, 73, 74, 76, 90. Both give minimum 10, $Q_1 = 25$, median 50, $Q_3 = 75$, maximum 90, an IQR of 50, no outliers, and a mean of 50, so the two boxplots are identical. Set A is spread evenly. Set B is two tight clumps with a hole through the middle, which a dotplot shows at a glance. A boxplot says nothing about where values sit inside a quarter, so it cannot show a second peak.

The other misreading appears whenever two boxplots share an axis: "group A's box sits entirely above group B's, so every student in A scored higher than every student in B." Take group A as 41, 62, 68, 72, 78, 82, 86, 95 and group B as 30, 44, 48, 55, 59, 61, 63, 85. A's box runs 65 to 84 and B's runs 46 to 62, so the two boxes never touch. B's top score of 85 still beats six of A's eight scores. A box covers the middle half of its group, not the group's range.

A boxplot also hides sample size completely: 8 observations and 800 with the same five numbers draw the same picture, so report $n$ beside it. When $n$ is small the five numbers nearly are the data, and a dotplot then shows everything the boxplot would and more. Boxplots are topic 1.8.