Quartile

$Q_1$, 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.