Interquartile Range
Compute the interquartile range for a set of data.
iqr(x, na.rm = FALSE)
x |
numeric vector of observations. |
na.rm |
logical scalar indicating whether to remove missing values from |
Let \underline{x} denote a random sample of n observations from some distribution associated with a random variable X. The sample interquartile range is defined as:
IQR = \hat{X}_{0.75} - \hat{X}_{0.25} \;\;\;\;\;\; (1)
where X_p denotes the p'th quantile of the distribution and \hat{X}_p denotes the estimate of this quantile (i.e., the sample p'th quantile).
See the R help file for quantile for information on how sample
quantiles are computed.
A numeric scalar – the interquartile range.
The interquartile range is a robust estimate of the spread of the
distribution. It is the distance between the two ends of a boxplot
(see the R help file for boxplot). For a normal distribution
with standard deviation σ it can be shown that:
IQR = 1.34898 σ \;\;\;\;\;\; (2)
Steven P. Millard (EnvStats@ProbStatInfo.com)
Chambers, J.M., W.S. Cleveland, B. Kleiner, and P.A. Tukey. (1983). Graphical Methods for Data Analysis. Duxbury Press, Boston, MA.
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Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY.
Hirsch, R.M., D.R. Helsel, T.A. Cohn, and E.J. Gilroy. (1993). Statistical Analysis of Hydrologic Data. In: Maidment, D.R., ed. Handbook of Hydrology. McGraw-Hill, New York, Chapter 17, pp.5–7.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.
# Generate 20 observations from a normal distribution with parameters # mean=10 and sd=2, and compute the standard deviation and # interquartile range. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rnorm(20, mean=10, sd=2) sd(dat) #[1] 1.180226 iqr(dat) #[1] 1.489932 #---------- # Repeat the last example, but add a couple of large "outliers" to the # data. Note that the estimated standard deviation is greatly affected # by the outliers, while the interquartile range is not. summaryStats(dat, quartiles = TRUE) # N Mean SD Median Min Max 1st Qu. 3rd Qu. #dat 20 9.8612 1.1802 9.6978 7.6042 11.8756 9.1618 10.6517 new.dat <- c(dat, 20, 50) sd(dat) #[1] 1.180226 sd(new.dat) #[1] 8.79796 iqr(dat) #[1] 1.489932 iqr(new.dat) #[1] 1.851472 #---------- # Clean up rm(dat, new.dat)
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