Median Absolute Deviation

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. This function wraps the specific base R function mad and extends it for the use of weights.

MAD(x, weights = NULL, center = Median, constant = 1.4826, 
    na.rm = FALSE, low = FALSE, high = FALSE)

Arguments

x

a numeric vector.

weights

a numerical vector of weights the same length as x giving the weights to use for elements of x.

center

the centre given either as numeric value or as a function to be applied to x (defaults to the DescTools::Median(x)). Note in cases when weights are defined to provide a function that also support weights. If this is not possible fall back to a numeric value.

constant

scale factor (default is 1.4826)

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, IQRw for weights, mad, median, var, MADCI (confidence intervals).

Examples

MAD(c(1:9))
#> [1] 2.9652
print(MAD(c(1:9),     constant = 1)) ==
      MAD(c(1:8, 100), constant = 1)       # = 2 ; TRUE
#> [1] 2
#> [1] TRUE
x <- c(1,2,3,5,7,8)
sort(abs(x - median(x)))
#> [1] 1 1 2 3 3 4
c(MAD(x, constant = 1),
  MAD(x, constant = 1, low = TRUE),
  MAD(x, constant = 1, high = TRUE))
#> [1] 2.5 2.0 3.0

# use weights
x <- sample(20, 30, replace = TRUE)
z <- as.numeric(names(w <- table(x)))

(m1 <- MAD(z, weights=w))
#> [1] 9.6369
(m2 <- MAD(x))
#> [1] 9.6369
stopifnot(identical(m1, m2))