Interquartile range

The interquartile range (IQR) is the difference between the 75th percentile (Q3) and the 25th percentile (Q1). It captures the middle 50% of a dataset — the range that contains the “typical” values.

For dataset [1, 3, 4, 5, 6, 7, 8, 9, 10, 20]: Q1 = 4, Q3 = 9, IQR = 5. The middle half of the data lies between 4 and 9. The outlier 20 doesn’t affect the IQR at all.

Why IQR matters: it’s robust against outliers in a way standard deviation isn’t. For skewed distributions (income, response times, file sizes), IQR describes spread far better than SD. The common 1.5 × IQR rule defines outliers as values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR — Tukey’s convention, and the basis of box plots.

Use IQR when: data is skewed, outliers are common, or you want a one-number summary that won’t mislead. Use standard deviation when: data is approximately normal and you want to feed it into downstream statistics (confidence intervals, regression).