stats (version 3.3.2)

mad: Median Absolute Deviation

Description

Compute the median absolute deviation, i.e., the (lo-/hi-) median of the absolute deviations from the median, and (by default) adjust by a factor for asymptotically normal consistency.

Usage

mad(x, center = median(x), constant = 1.4826, na.rm = FALSE,
    low = FALSE, high = FALSE)

Arguments

x

a numeric vector.

center

Optionally, the centre: defaults to the median.

constant

scale factor.

na.rm

if TRUE then NA values are stripped from x before computation takes place.

low

if TRUE, compute the ‘lo-median’, i.e., for even sample size, do not average the two middle values, but take the smaller one.

high

if TRUE, compute the ‘hi-median’, i.e., take the larger of the two middle values for even sample size.

Details

The actual value calculated is constant * cMedian(abs(x - center)) with the default value of center being median(x), and cMedian being the usual, the ‘low’ or ‘high’ median, see the arguments description for low and high above. The default constant = 1.4826 (approximately \(1/\Phi^{-1}(\frac 3 4)\) = 1/qnorm(3/4)) ensures consistency, i.e., $$E[mad(X_1,\dots,X_n)] = \sigma$$ for \(X_i\) distributed as \(N(\mu, \sigma^2)\) and large \(n\). If na.rm is TRUE then NA values are stripped from x before computation takes place. If this is not done then an NA value in x will cause mad to return NA.

See Also

IQR which is simpler but less robust, median, var.

Examples

mad(c(1:9))
print(mad(c(1:9),     constant = 1)) ==
      mad(c(1:8, 100), constant = 1)       # = 2 ; TRUE
x <- c(1,2,3,5,7,8)
sort(abs(x - median(x)))
c(mad(x, constant = 1),
  mad(x, constant = 1, low = TRUE),
  mad(x, constant = 1, high = TRUE))