Quartiles and the interquartile range:
splitting data and spotting outliers

What the three quartiles actually split

Sort a data set from smallest to largest and cut it into four equal-sized parts. The three cut points are the quartiles. The middle cut is the median, Q2. Q1, the lower quartile, has a quarter of the values below it. Q3, the upper quartile, has a quarter above it.

The stretch from Q1 to Q3 holds the central half of the data. That distance is the interquartile range, IQR = Q3 − Q1. Because it ignores the lowest and highest quarters, one wild value hardly moves it, which makes it a sturdier measure of spread than the range or the standard deviation.

These numbers underpin the box-and-whisker plot: the box runs from Q1 to Q3, a line marks the median, and whiskers extend to the most extreme values that are not flagged as outliers.

Finding Q1 and Q3 step by step

1. 1Sort the values in ascending order.
2. 2Find the median. If n is odd, it is the middle value; if even, the average of the two middle values.
3. 3Split the data into a lower half and an upper half. For odd n, the median is either left out of both halves or included, depending on the method.
4. 4Q1 is the median of the lower half; Q3 is the median of the upper half.
5. 5Subtract: IQR = Q3 − Q1.

With an even count the halves are clean. With an odd count textbooks differ about the middle value, which is why different tools can return slightly different quartiles for the same list.

A worked example with twelve values

Here are the ages in months at which twelve toddlers in a play group took their first steps, already sorted: 7, 9, 12, 13, 15, 16, 18, 21, 24, 26, 29, 58. With 12 values the lower half is the first six and the upper half is the last six.

Note how the 58 sits far out on its own. The median and quartiles barely notice it; the mean of this list, about 20.7, is pulled upward by it. That asymmetry is exactly what the next step is designed to formalise.

An odd count: what happens to the middle value

With nine values the median is a data point itself, which forces a choice. Take 3, 5, 6, 8, 9, 11, 12, 15 and 40.

Neither rule is wrong. What matters is to say which one you used, and to compare data sets only when they were treated the same way.

The 1.5 × IQR rule for flagging outliers

A widely used convention, attributed to the statistician John Tukey, builds two fences outside the box. A value below the lower fence or above the upper fence is marked as a potential outlier.

'Potential' is the word to hold on to. The rule is a screening device, not a verdict. A flagged value might be a typing error, a measurement fault, or a perfectly genuine rare observation; it is worth checking the source before deleting it.

Why two tools can give different quartiles

There is no single agreed way to place quartiles when the cut falls between two data points. Common methods give slightly different answers, and all are legitimate if stated.

All three agree that 58 is an outlier, which is the usual outcome: methods differ in the second decimal, not in the story. Pick one method, note it, and stay with it when comparing data sets.

Reading a box plot with these numbers

A box plot draws the five-number summary: minimum, Q1, median, Q3 and maximum. The box spans Q1 to Q3, the line inside marks the median, and the whiskers reach the smallest and largest values that are inside the fences. Points beyond the fences are drawn separately as dots.

Two features are worth a glance. First, the position of the median inside the box: if it sits near one end, the data is skewed that way. Second, whisker lengths: a long upper whisker and a short lower one signal a tail of high values.

Q1 is the 25th percentile and Q3 the 75th, so a score at the 75th percentile is at Q3: three-quarters of the group scored the same or lower.

When to prefer the IQR to the standard deviation

• The data is skewed, such as incomes, house prices or response times, where a few huge values inflate the standard deviation.
• There are outliers you cannot remove or have not yet investigated.
• You are summarising with a median; the IQR pairs naturally with it, as the standard deviation pairs with the mean.
• The distribution is ordinal or the exact values are uncertain, but the ranking is reliable.

For roughly symmetric, well-behaved data the standard deviation carries more information. When in doubt, report both and look at the gap: a standard deviation far larger than the IQR ÷ 1.35 hints at a heavy tail.