5.5 : Quartile

Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first, find the median or second quartile. The first quartile, Q₁, is the middle value of the lower half of the data, and the third quartile, Q₃, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:

1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5

The median or second quartile is seven. The lower half of the data are 1, 1, 2, 2, 4, 6, and 6.8. The middle value of the lower half is two. 1; 1; 2; 2; 4; 6; 6.8. The number two, which is part of the data, is the first quartile. One-fourth of the entire sets of values are the same as or less than two, and three-fourths are more than two. The upper half of the data is 7.2, 8, 8.3, 9, 10, 10, and 11.5. The middle value of the upper half is nine.

The third quartile, Q₃, is nine. Three-fourths (75%) of the ordered dataset is less than nine. One-fourth (25%) of the ordered dataset is greater than nine. The third quartile is part of the dataset in this example.

The interquartile range is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile (Q₃) and the first quartile (Q₁).

IQR = Q₃ – Q₁

The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile.

This text is adapted from Openstax, Introductory Statistics, Section 2.3 Measures of the Location of the Data