The MAD function computes the univariate (scaled) median absolute deviation of each column of the input matrix.

The arguments to the MAD function are as follows:

x

is an input data matrix.

method

is an optional string argument with the following values:

"MAD"

for computing the median absolute deviation (MAD); this is the default.

"NMAD"

for computing the normalized version of MAD

"SN"

for computing

"QN"

for computing

For simplicity, the following descriptions assume that the input argument x is a column vector. The notation means the th element of the column vector x.

The MAD function can be used for computing one of the following three robust scale estimates:

• median absolute deviation (MAD) or normalized form of MAD,

  where is the unscaled default and is used for the scaled version (consistency with the Gaussian distribution).

• , which is a more efficient alternative to MAD,

  where the outer median is a low median (order statistic of rank ) and the inner median is a high median (order statistic of rank ), and where is a scalar that depends on sample size .

• is another efficient alternative to MAD. It is based on the th-order statistic of the inter-point distances,

  where is a scalar similar to but different from . See Rousseeuw and Croux (1993) for more details.

The scalars and are defined as follows:

 b = {3, 4, 7, 8, 10, 949, 951};

 rmad1 = mad(b);
 rmad2 = mad(b,"mad");
 rmad3 = mad(b,"nmad");
 rmad4 = mad(b,"sn");
 rmad5 = mad(b,"qn");
 print "Default MAD=" rmad1,
       "Common MAD =" rmad2,
       "MAD*1.4826 =" rmad3,
       "Robust S_n =" rmad4,
       "Robust Q_n =" rmad5;