IQR() — Interquartile range

How wide the middle half of the numbers is.

IQR(x)

Key idea

• The IQR is Q3 (75th percentile) minus Q1 (25th percentile): the spread of the middle 50% of the data. (e.g. Q1=10, Q3=18 -> IQR 8)
• It ignores the lowest and highest quarters, so outliers (far-off values) do not affect it.
• A small IQR means the central values are tightly clustered together.
• The IQR is often used to flag outliers beyond 1.5x IQR from the quartiles. (e.g. 1.5 x 8 = 12)
• It pairs well with the median (the middle value) to summarize skewed data.

What are quartiles?

1. Sort the data from smallest to largest. Quartiles are three cut points — Q1, Q2, Q3 — that split the sorted data into four equal quarters, 25% of the values in each.
2. Q1, the lower (first) quartile, has 25% of the values below it. It is roughly the median of the lower half — e.g. the 25th percentile. (Software often interpolates, so the exact value can differ slightly.)
3. Q2 is just the median itself: 50% of the values fall below it. It splits the data into two halves.
4. Q3, the upper (third) quartile, has 75% of the values below it. It is roughly the median of the upper half — e.g. the 75th percentile.
5. The IQR is the gap between Q1 and Q3 (IQR = Q3 − Q1) — the width of the middle 50%, and exactly the length of the box in a boxplot.

Definition

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

$Q_1$

the 25th percentile (lower quartile)

$Q_3$

the 75th percentile (upper quartile)

$$\mathrm{IQR}=Q_3-Q_1=7-3=4$$

Where you'd use it

• House prices on a street run from $180k up to a $2M mansion. If the middle half sits between $220k and $300k, the IQR is $80k, a steady measure of spread that the lone mansion cannot stretch.
• Salaries in a firm have a few huge boss pays. With the first quartile at $35k and the third at $55k, IQR = $20k captures the typical pay spread while ignoring the extremes.
• Exam scores have first quartile 62 and third quartile 81. The IQR of 19 points tells you how wide the central half of the class scored, the box length you would see on a boxplot.

Frequently asked questions

What is the IQR?

Q3 − Q1, the spread of the middle 50% of the data — robust to outliers.