Interquartile Range

The interquartile range (IQR) is a resistant measure of spread for a quantitative data set. A ‘resistant measure’ is not influenced by outliers. A ‘measure of spread’ indicates how consistent or variable the data set is. Other measures of spread that are not resistant to outliers are the standard deviation and the range (maximum – minimum). The IQR measures the number of units in which the middle 50% of the data lie and is used in one technique to determine outliers in a data set.

The IQR is the difference between the third and first quartiles, Q3 and Q1, respectively, that is, IQR = Q3 − Q1. The IQR is always a positive value. The Q3 value is the 75th percentile, while Q1 is the 25th percentile; hence, the difference between these values gives the distance that contains the middle 50% of the observations. Different statistical packages may calculate Q1 and Q3 differently, which leads to slightly different values of the IQR. The difference in values between statistical packages is usually insignificant.

Consider Data Sets A and B in Tables 1 and 2, respectively, (created by the author for this entry).

By observation, one sees that Data Set B is more consistent than Data Set A. Data Set A is more variable. The first and third quartiles for the data sets are Q1A = 5, Q3A = 55, Q1B = 112, and Q3B = 117. Hence, IQRA = 55 − 5 = 50 and IQRB = 117 − 112 =5. The middle 50% of the observations for Data Set A are contained within 50 units, while for Data [Page 558]Set B it only takes 5 units. Since IQRA > IQRB, Data Set A is more variable than Data Set A.

One can observe that any changes in the minimums or maximums in the data sets will not affect the quartiles, which means that the IQR will not be affected. Hence, the IQR is resistant to outliers.

Outliers and the IQR

The IQR is also used to determine outliers in a data set. The rule of thumb is this: If an observation is 1.5 IQRs away from either the first or third quartile, then the observation is considered an outlier.

Determining whether Data Set A contains any outliers:

Since Data Set A does not contain any values below the lower fence or above the upper fence, Data Set A does not contain any outliers.

Determining whether Data Set B contains any outliers:

Data Set B does not have any value below the lower fence, but the value 130 is above the upper fence. Hence the value 130 is considered an outlier.

A data set may have multiple outliers.