Imagine lining up all 32 runners in a race by their finish times, then splitting the line into four equal groups: the fastest quarter, the next quarter, the next, and the slowest quarter. The three points where you cut are the quartiles. They give you a robust, at-a-glance picture of where the middle of your data lives — and how tightly it clusters — without a single wild value being able to throw the whole story off.

The median already splits sorted data into two halves. Quartiles go one step further and split it into four equal quarters using three cut points:

• Q_1 — the lower quartile: a quarter of the data lie below it.
• Q_2 — the median: half lie below it.
• Q_3 — the upper quartile: three quarters lie below it.

Here is the trick that makes them easy to find: each quartile is itself a median. Q_2 is the median of the whole set; Q_1 is the median of the lower half; Q_3 the median of the upper half. If you can find a median, you can find all three quartiles.

Worked example: finding all three quartiles

Take eight sorted values — say the number of books eight friends read last month:

There are 8 values, an even count, so the median is the average of the 4th and 5th: Q_2 = \tfrac{7 + 8}{2} = 7.5. That splits the data into a lower half \{2, 4, 5, 7\} and an upper half \{8, 11, 12, 16\}.

• Lower half median: Q_1 = \tfrac{4 + 5}{2} = 4.5.
• Upper half median: Q_3 = \tfrac{11 + 12}{2} = 11.5.

So the middle half of the data runs from 4.5 up to 11.5.

The interquartile range

The distance between the outer quartiles is the interquartile range:

It is the width of the middle 50% of the data. For the books example above, \text{IQR} = 11.5 - 4.5 = 7 books. Unlike the range, which is decided entirely by the two most extreme points, the IQR throws those extremes away. That makes it robust: a single wild outlier can blow up the range but barely nudges the IQR.

Worked example: why "robust" matters

Nine houses on a street have these prices (in hundreds of thousands):

Here the range is 7 - 3 = 4, and the quartiles are Q_1 = 4, Q_3 = 6, so \text{IQR} = 2. Now a mansion sells and the last value jumps from 7 to 90:

• The range explodes from 4 to 90 - 3 = 87 — more than twentyfold.
• The standard deviation similarly balloons, because it squares that huge gap.
• The IQR stays at 2 (the quartiles Q_1 = 4 and Q_3 = 6 haven't budged) — the outlier is out past the whisker where the IQR simply doesn't look.

One point changed the range by a factor of twenty and left the IQR untouched. That is exactly why analysts reach for the IQR when a data set might contain a rogue value.

The 1.5 × IQR rule for outliers

Because the IQR ignores extremes, it makes a great ruler for spotting them. The classic rule builds two "fences" one and a half IQRs beyond the box:

Any value beyond a fence is flagged as a suspected outlier. Back to the book example, where Q_1 = 4.5, Q_3 = 11.5 and \text{IQR} = 7:

So a friend who read 16 books is not an outlier (it sits below 22), but a friend who read 30 would be flagged. The 1.5 isn't sacred — it is Tukey's convenient default that catches genuinely stray points without crying wolf over ordinary variation.

The boxplot

A boxplot draws all five summary numbers at once. For the sorted data set 3, 5, 6, 7, 8, 9, 11, 13, 15, 21 the quartiles are Q_1 = 6, median = 8.5 and Q_3 = 13, so \text{IQR} = 13 - 6 = 7. The box runs from Q_1 to Q_3 with the median marked inside; the whiskers reach out to the smallest and largest values.

At a glance the box shows where the central half of the data sits, and the whiskers show how far the tails stretch. A long whisker on one side already hints at skew.

See it explained