quartile

A quartile is a type of quantile that divides a ranked data set into four equal parts. The three quartiles, Q1, Q2, and Q3, correspond to the 25th, 50th, and 75th percentiles respectively. Q2 is commonly the median of the data.

Q1 is the value such that about a quarter of observations fall at or below it; Q3 is the value such that about three quarters fall at or below it. Together, Q1, Q2, and Q3 summarize the central tendency and the spread of the data, and they are used in the construction of box plots and in robust descriptive statistics.

Calculation: several conventions exist. One widely used method defines Q2 as the median of the data; Q1 as the median of the lower half and Q3 as the median of the upper half (the halves are formed by splitting the data at the median). An alternative approach uses linear interpolation to estimate quartiles from the ordered data, assigning the p-th quartile to the value at position (n+1)p/4, with interpolation for fractional positions.

Interquartile range (IQR) is Q3 minus Q1. It measures the spread of the middle 50% of the data. Outliers are often identified as observations below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

In practice, quartiles provide a simple, robust summary for skewed distributions and are commonly reported alongside the mean. They are widely supported in statistics software; however, different definitions can yield slightly different quartile values, especially for small samples.