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Quartiles and IQR

Splitting data into quarters to summarise the middle half without letting extremes rule.

Data Analyst · Mid-level

Also known as: IQR, interquartile range, quartiles

Quartiles cut a sorted dataset into four equal parts. The first quartile, Q1, is the value below which a quarter of the data falls; the median is Q2; the third quartile, Q3, is the value below which three quarters falls. The interquartile range is Q3 − Q1 — the width of the middle half of your data.

That is what makes it useful when there are extremes in the column. For a set of order values that sit around 15 with one at 4,000, the IQR does not move, while the range and the standard deviation both do. It is the same reason a box plot draws its box from Q1 to Q3 and leaves the far points outside it.

Three things to get right:

  • Quartiles are percentiles, and percentile conventions differ by tool. There is no single answer to “the 25th percentile of these eight numbers” when the cut lands between two rows. Excel’s QUARTILE.INC interpolates and QUARTILE.EXC uses a different position; numpy’s np.percentile and the Python standard library’s quantiles() default to different interpolation methods; SQL engines distinguish PERCENTILE_CONT from PERCENTILE_DISC. The differences are usually small and shrink as n grows, but a dashboard disagreeing with a notebook by a hair is often this. Pick the convention your reporting uses and stay with it.
  • Quartiles of small samples are noisy. A Q1 from twelve rows is mostly a statement about two of them.
  • You cannot average quartiles. Combining two groups by averaging their Q1 does not give the Q1 of the pooled data. Recompute it from the underlying rows, as with any percentile.

Outliers are commonly flagged with the fence Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. The 1.5 is a convention rather than a discovery — it flags rows for inspection, it does not condemn them. See outliers for what to do next.

The trade-off against the standard deviation is that the IQR throws away information by using only two values, so it is noisier on well-behaved data. It earns its keep on data that is not: skewed columns, long-tailed metrics, and anything with a handful of extreme rows. Report both when the shape is unclear, and use descriptive statistics as the frame.