Quartile

Finding them

Sort the data and find the median. Then:

• $Q_1$ is the median of the values below the median.
• $Q_3$ is the median of the values above it.

With an odd count, leave the median itself out of both halves.

For $4, 7, 9, 12, 14, 15, 21$ the median is $12$:

$$\text{lower } (4, 7, 9) \Rightarrow Q_1 = 7 \qquad \text{upper } (14, 15, 21) \Rightarrow Q_3 = 15$$

What each one marks

	Values below it
$Q_1$	$25\%$
median	$50\%$
$Q_3$	$75\%$

So a test score at $Q_3$ beat three quarters of the group.

The interquartile range

$$\text{IQR} = Q_3 - Q_1$$

The IQR is the width of the middle half of the data. Because it discards the top and bottom quarters, an outlier sitting in one of those tails has no effect on it — which is what makes it the measure of spread to pair with a median.

On a box plot, the box runs from $Q_1$ to $Q_3$, so the width of the box is the IQR.

A note on conventions

Statistical software offers several methods for computing quartiles, and they disagree on small data sets. The method above — splitting at the median and excluding it when the count is odd — is the one US school courses use, and it is what this site uses throughout.