Quartiles

Quartiles are values that divide a rank-ordered data set into four equal-sized groups. They are commonly denoted as Q1, Q2, and Q3, corresponding to the 25th, 50th, and 75th percentiles, respectively. Q2 is the median of the data; Q1 marks the median of the lower half, and Q3 marks the median of the upper half. Because several conventions exist for calculating quartiles, especially for small data sets, the exact numerical values can differ slightly across methods.

For samples, a widely used approach is to order the data and take Q2 as the overall median, Q1 as the median of the lower half (values below the median), and Q3 as the median of the upper half (values above the median). In data sets with an even number of observations, the lower and upper halves are disjoint; with an odd number, the median is typically excluded from both halves.

The interquartile range, IQR, equals Q3 minus Q1 and measures the spread of the middle 50% of the data. Quartiles are a fundamental tool in descriptive statistics and are often reported to summarize distribution shape and variability. They relate directly to percentiles, with Q1 as the 25th percentile, Q2 as the 50th percentile (the median), and Q3 as the 75th percentile. Quartiles also support outlier detection, where observations below Q1 − 1.5×IQR or above Q3 + 1.5×IQR may be considered potential outliers. In population analyses, quartiles partition the population into four equal-probability intervals.