## 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, 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.

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

### Examples

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

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