Quartile vs Percentile

$Q_1$, 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 $k$th percentile is the value at or below which about $k$ percent of the observations lie. The grammar of that sentence is worth slowing down for: $k$ is the percent, and the percentile is the value, carrying whatever units the data carry. If the 90th percentile of a test is 628 points, then 90 percent of scores are 628 or lower and about 10 percent are higher.

Run the definition backwards and you get the percentile rank, which starts from a value and returns a percent:

Take the ten scores 52, 58, 60, 64, 65, 69, 73, 79, 86, 91. Seven of them are 73 or lower, so the percentile rank of 73 is $\frac{7}{10} \times 100 = 70$, and 73 is the 70th percentile of this set. Move the same score into a different group and its percentile rank changes, because a percentile always describes a position within one particular set of data.

Percentiles work on a smooth distribution too, where they are read off the curve instead of counted. On a normal distribution with mean 500 and standard deviation 100, the 90th percentile sits about 1.28 standard deviations above the mean, at $500 + 1.28(100) = 628$ points. That is the same 628 from the opening paragraph, and it shows the two routes to a percentile: counting observations in a data set, or finding the area to the left under a density curve.

The misreading that costs the most marks is treating a percentile as a score. Standing at the 85th percentile is a claim about the other test-takers, not about how many questions you answered correctly. On a hard test you can get 40 percent of the items right and still sit at the 85th percentile of it.

Three percentiles have names of their own: the 25th is the first quartile $Q_1$, the 50th is the median, and the 75th is the third quartile $Q_3$. Percentile and z-score are the two standard ways of reporting relative standing, and they answer different questions. A percentile says how many values sit at or below yours; a z-score says how far above or below the mean yours is, measured in standard deviations.