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 MAD (which 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 .

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 depending 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;