Interquartile Range

Descriptive statistics summarize or describe data sets using measures of central tendency and dispersion. Measures of central tendency (e.g., mean, median) identify the dominant, representative, or typical data value, whereas measures of dispersion (e.g., standard deviation) communicate the spread or variability in the data set. Interquartile range (IQR) is a measure of dispersion that encompasses the middle half of the data by taking the difference between the data values positioned at the 25th and 75th percentiles. The IQR accentuates the central range of the data rather than the maximum and minimum values. This entry explains [Page 862]how to calculate the IQR as well as how IQRs are commonly displayed graphically.

To determine the IQR, the data are first arranged in ascending order and subdivided into four equal portions or quartiles. Each quartile contains 25% of the data observations. Next, the data values associated with the 25th and 75th percentiles are determined. For n observations, the 25th percentile or first quartile (Q1) data value occurs at (n + 1)/4 and the 75th percentile or third quartile (Q3) data value occurs at 3(n + 1)/4; the 50th percentile or second quartile is the median. Often whole integers do not result and interpolation is required. For example, a data set with 16 observations denotes that Q1 occurs at the 4.25 observation, signifying that Q1 is the fourth observation plus 0.25 times the difference between the values of the fourth and fifth observations. Finally, the IQR is found by subtracting the Q1 data value from the Q3 data value. A large (small) IQR indicates a data set with a greater (lesser) central dispersion and more (less) variability.

The following data set (n = 11), prearranged in ascending order, is used to illustrate the process for obtaining the IQR:

Based on the quartile data position formulas, Q1 occurs at the third observation (9) and Q3 occurs at the ninth observation (16); these values represent the median of the lower and upper portions of the data set, respectively. The IQR is the difference between Q3 and Q1 or 7. In comparison with the data range (i.e., the difference between the maximum and minimum values), the IQR in this example is relatively small and indicates less variability in the central half of the data set than with the data set as a whole.

The IQR is commonly displayed as part of a box plot, a type of graph that shows the position of Q1 and Q3, the median, and the data range. In a box plot, the rectangular box is created from the Q1 and Q3 boundaries, thus highlighting the IQR and the middle half of the observations. The median (second quartile) and sometimes the mean are displayed as single lines within the box. Lines or “whiskers” are drawn extending from the box edges in opposite directions until the maximum and minimum values that are not outlier values are reached. Outliers are defined as data points more (less) than 1.5 IQR above (below) the third (first) quartile; outliers are represented by a single identifier (e.g., asterisk) beyond the whiskers.

...