Five-Number Summary vs Quartile

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.

$Q_1$, the first quartile, is the median of the lower half of the sorted data, and $Q_3$, the third quartile, is the median of the upper half, with the overall median between them as the second quartile. Everything turns on what you do with that median when $n$ is odd. This site uses the median-excluded convention, the one TI-83 and TI-84 calculators use: the median is left out of both halves before you take the median of each.

Take the seven values 4, 6, 9, 11, 15, 18, 20. The median is the fourth value, 11. Median-excluded leaves a lower half of 4, 6, 9 and an upper half of 15, 18, 20, so $Q_1 = 6$, $Q_3 = 18$, and the IQR is 12. The median-included convention keeps the 11 in both halves: 4, 6, 9, 11 gives $Q_1 = (6 + 9)/2 = 7.5$, and 11, 15, 18, 20 gives $Q_3 = 16.5$, for an IQR of 9. Same seven numbers, different quartiles, neither an arithmetic error.

So "my calculator says $Q_1 = 6$ but the spreadsheet says 7.5, so one of them is wrong" is itself the mistake. There is no universal definition of a quartile in a finite data set, software disagrees on purpose, and naming the convention you used ends the argument. The two rules always agree when $n$ is even, since there is no middle value to place, so the question arises only for odd $n$.

A quartile is a value, not a group. "The first quartile is the bottom 25 percent of the scores" describes a set of observations; $Q_1 = 6$ is one number in the units of the data. The four groups are only roughly equal too: with $n = 7$ no cut gives four of the same size.

The choice propagates: the IQR is $Q_3 - Q_1$ and the 1.5 IQR fences are built from it, so switching conventions can move a borderline value from inside a fence to outside. In the Fall 2026 course this sits in Unit 1, whose topic 1.7 is titled Summary Statistics for One Quantitative Variable.