Quartile vs IQR

$Q_1$, 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.

The IQR is one number: $Q_3 - Q_1$, the third quartile minus the first. It carries the units of the data and measures the span the middle 50 percent of the values occupy. Because the top and bottom quarters are discarded before anything is measured, no value out in either tail can touch it. The IQR is the spread you quote beside the median, as the standard deviation goes beside the mean.

For 3, 6, 7, 9, 12, 14, 15, 40 the median is $(9 + 12)/2 = 10.5$. The lower half is 3, 6, 7, 9, giving $Q_1 = 6.5$; the upper half is 12, 14, 15, 40, giving $Q_3 = 14.5$; so the IQR is $14.5 - 6.5 = 8$. Change the 40 to 400 and the IQR is still 8, while the range grows from 37 to 397 and the sample standard deviation goes from about 11.6 to 138. That is what resistance looks like in numbers.

The IQR is a width, not a location, and writing it as one is the standard error. "The IQR is from 6.5 to 14.5" reports the interval the middle half occupies; the IQR is the single number 8. The same slip appears as "the IQR is the middle 50 percent of the data," which names observations rather than a measurement of them.

The width says nothing about where the median sits inside it. In 1, 2, 3, 4, 5, 6, 20, 30, 40 the quartiles are 2.5 and 25, an IQR of 22.5, while the median is 5, pressed against the left edge of that wide middle half.

One edge case breaks the usual outlier rule. If more than half the values are identical, $Q_1$ and $Q_3$ can coincide. For 5, 5, 5, 5, 5, 5, 5, 5, 9, 20 both quartiles are 5, the IQR is 0, and both 1.5 IQR fences collapse onto 5, so the rule flags every value other than 5, including the 9. The arithmetic is right and the conclusion is useless, so say so rather than reporting outliers with a straight face.