Interquartile Range (IQR) Calculator

The IQR measures the spread of the middle 50% of the data.

How the Math Works

The Interquartile Range (IQR) is calculated by first sorting your data in ascending order, then determining the first quartile (Q1) and third quartile (Q3). Q1 represents the 25th percentile - the point below which 25% of your data falls, while Q3 represents the 75th percentile - the point below which 75% of your data falls. The IQR formula is elegantly simple: IQR = Q3 - Q1. This subtraction gives you the range that contains the middle 50% of your data, effectively measuring the spread while ignoring extreme outliers. Different methods exist for calculating quartiles, but they all converge on this fundamental principle of subtracting the first quartile from the third quartile to reveal the core variability in your dataset.

Practical Applications

To apply this calculation practically, start by organizing your data points from smallest to largest. For example, if analyzing test scores of 12 students: 65, 70, 72, 78, 80, 82, 85, 88, 90, 92, 95, 98, you would find Q1 (the median of the lower half) at 75 and Q3 (the median of the upper half) at 90, giving an IQR of 15. This calculation is invaluable for identifying outliers using the 1.5×IQR rule: any values below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR) are considered statistical outliers. Researchers use IQR in quality control to detect anomalies, in finance to measure stock price volatility, and in education to understand score distributions and identify students who may need additional support or enrichment.

Day-to-Day Use

In everyday life, the IQR helps you make sense of numerical information presented in news articles, reports, and data visualizations without getting overwhelmed by extreme values. When a weather report mentions that typical daily temperatures have an IQR of 8 degrees, you understand the normal variation in temperature swings. If a salary comparison shows an IQR of $15,000 for a job position, you have realistic expectations about compensation ranges rather than being misled by unusually high or low outlier salaries. This measure is particularly useful when evaluating product reviews, where an IQR can tell you whether most customers have similar experiences or if experiences vary wildly. Understanding IQR empowers you to critically assess data-driven claims and make more informed decisions based on the typical, rather than exceptional, circumstances.

Worked example

4,8,15,16,23 → Q1 8, Q3 16, IQR 8.

FAQ