Quartile

A descriptive statistic dividing a ranked dataset into four equal-sized groups, marking the twenty-fifth, fiftieth, and seventy-fifth percentiles.

Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.

A quartile divides a ranked dataset into four equal parts, marking the points below which a quarter, a half, and three-quarters of the observations fall. Splitting the data this way shows how it is distributed, locating the middle and the spread of the bulk of values, and the gap between the first and third quartiles gives the interquartile range. Quartiles are simply the twenty-fifth, fiftieth, and seventy-fifth percentiles. Carving a distribution into four equal parts is what it does. Kirkwood and Sterne (2003) describe this.

Source: Kirkwood & Sterne 2003

Quartiles are calculated by ordering the data and finding the values that divide it into four equal parts: the first quartile is the value below which a quarter of the data lie, the second is the median, and the third is the value below which three quarters lie, with interpolation used when positions fall between data points. So quartiles are calculated from the ordered data by locating these dividing points, and because different conventions exist for interpolation, the exact method can vary slightly, though the quartiles reliably mark the quarter, half, and three-quarter positions of the distribution, providing the basis for the interquartile range and box plots.

Source: Casella & Berger 2002

Quartiles are used to describe the spread and position of data, to compute the interquartile range as a robust measure of dispersion, and to construct box plots that display the median, quartiles, and outliers. So quartiles are used to summarise a distribution robustly, since they are based on rank and resistant to outliers, which makes them valuable for skewed data, and they underpin common descriptive tools, providing a clear picture of where the data are concentrated and how spread out they are through the middle half, complementing the median as a summary of the centre.

Source: Casella & Berger 2002

Quartiles relate to the interquartile range in that the interquartile range is the difference between the third and first quartiles, that is the spread of the middle half of the data. The quartiles define the boundaries of this central portion. So quartiles and the interquartile range are directly linked, with the first and third quartiles marking the range that contains the central fifty per cent of the data and the interquartile range measuring its width, which is why quartiles are the basis for this robust measure of dispersion, resistant to outliers because it ignores the extreme quarters of the data.

Source: Casella & Berger 2002

Quartiles relate to percentiles as specific percentiles: the first quartile is the twenty-fifth percentile, the second, or median, is the fiftieth, and the third is the seventy-fifth. Quartiles thus divide the data into four parts, while percentiles allow division into a hundred. So quartiles are particular percentiles marking the quarter points of a distribution, part of the broader family of quantiles, which is why they can be expressed as percentiles, and the choice between quartiles, deciles, and percentiles depends on how finely the distribution's positions need to be described for the purpose at hand.

Source: Casella & Berger 2002