Quartile

Quartile is a rank-order grouping. A quartile divides a distribution of data into four equally sized groups determined after ranking the data according to some measure or combination of measures.

[Page 1353]There is some debate about who first used the term quartile (contenders include Carl F. Gauss, Donald McAlister, and Francis Galton), but it seems that Galton was the first to bring the terms quartile, decile, and percentile into common use.

The term quartile can have two meanings: a point on the distribution that divides the four groups, and the group to which a member of the distribution belongs. There are three decile values, which separate the data into the four groups. The third decile value (commonly known as the upper quartile) is the point in the distribution where three quarters of the data lies below that value. The second decile (commonly known as the median) is the point below which two quarters (one-half) of the distribution lie. One quarter of the distribution lies below the first quartile (known as the lower quartile). If a point on the distribution lies between the median and upper quartile, then it is a member of quartile 3. The interquartile range is the difference between the upper quartile and the lower quartile.

There is no precise definition for calculating the quartile values. Different software and different statisticians can use different values for the same distribution, but the differences are usually not important enough to impact the exploratory nature of this description or the interpretation of the data.

When an analyst breaks data into quartile groups, the purpose is to simplify the way in which they can describe and visualize the data, such as to look for patterns and trends, or to compare and contrast high-performing and low-performing groups.

A good example of this is the five-point summary, and the visual representation of this summary, the box plot. Box plots (or box-and-whisker plots) were devised by John Tukey in 1969 as an exploratory data tool to visualize some characteristics of a distribution. The box is plotted using the median and the upper and lower quartiles (called hinges by Tukey). Therefore, half of the data lies inside the box. There are variations in the statistic used to determine the length of the whiskers. Tukey used 1.5 × interquartile range to determine the end of the whiskers (he called these the inner fences), but the maximum and minimum value are also commonly used. There are other less common variations, so it is important to state what convention has been used. If values other than the maximum and minimum are used for the whiskers, outliers can be also shown.

See also Box Plot; Decile; Descriptive Statistics; Interquartile Range; Percentile Rank