Iqr

IQR, or interquartile range, is a measure of statistical dispersion, describing the spread of the middle 50 % of a data set. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1), where Q1 marks the 25th percentile and Q3 marks the 75th percentile. By focusing on the central portion of the distribution, the IQR reduces the influence of extreme values and outliers, making it a robust alternative to the total range.

To compute the IQR, a data set is first ordered from smallest to largest. The position of Q1 is typically determined as the median of the lower half of the data, while Q3 is the median of the upper half. Various interpolation methods exist for data sets with sizes that do not allow exact quartile positions, but the principle remains the same: IQR = Q3 − Q1.

The IQR is commonly used in exploratory data analysis, box plots, and as a criterion for detecting outliers. A standard rule identifies points lying more than 1.5 × IQR below Q1 or above Q3 as potential outliers. Because it summarises variability without being swayed by extreme scores, the IQR is valuable for comparing distributions with differing shapes or scales.

In addition to its descriptive role, the IQR assists in non‑parametric statistical methods, such as the Kruskal‑Wallis test, and provides a basis for constructing confidence intervals for medians. Its simplicity and resistance to outliers make the interquartile range a fundamental tool in both academic research and practical data analysis.