interquartilerange

Interquartile range

The interquartile range (IQR), also known as the middle fifty, is a measure of statistical dispersion that describes the spread of the central half of a dataset. It is defined as the difference between the third quartile (Q3, the 75th percentile) and the first quartile (Q1, the 25th percentile).

Calculation and quartiles: To compute the IQR, order the data from smallest to largest. Q1 is the value below which 25% of observations fall, and Q3 is the value below which 75% fall. The IQR = Q3 − Q1. There are several conventions for determining quartiles, especially for small samples or when using interpolation; different software packages may yield slightly different Q1 and Q3 values, but the IQR remains a robust central-dispersion measure.

Interpretation: The IQR measures the spread of the middle 50% of values and is less influenced by extreme observations than the full range. A larger IQR indicates greater variability in the central portion of the data.

Applications: Box plots commonly display the IQR as the height of the box, with whiskers extending to 1.5 times the IQR from the quartiles; data points beyond that range are often flagged as potential outliers. The IQR is widely used in descriptive statistics, nonparametric analyses, and robust data summaries.

Notes: The IQR does not reflect the shape of the entire distribution and ignores variation outside the central half. It is often used together with the median for a robust summary of a dataset.