Relation To Standard Deviation
In order to use the MAD as a consistent estimator for the estimation of the standard deviation σ, one takes
where K is a constant scale factor, which depends on the distribution.
For normally distributed data K is taken to be 1/Φ−1(3/4) 1.4826, where Φ−1 is the inverse of the cumulative distribution function for the standard normal distribution, i.e., the quantile function. This is because the MAD is given by:
Therefore we must have that Φ(MAD/σ) − Φ(−MAD/σ) = 1/2. Since Φ(−MAD/σ) = 1 − Φ(MAD/σ) we have that MAD/σ = Φ−1(3/4) from which we obtain the scale factor K = 1/Φ−1(3/4).
Hence
In other words, the expectation of 1.4826 times the MAD for large samples of normally distributed Xi is approximately equal to the population standard deviation.
The factor results from the reciprocal of the normal inverse cumulative distribution function, evaluated at probability .
Read more about this topic: Median Absolute Deviation
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