Objective
The objective of this webpage is to provide a proof of Property 3 of Measures of Variability.
Property 3: For a normal distribution,
σ = k ⋅ MAD
where k = 1/Φ-1(.75) = 1/NORM.S.INV(.75) ≈ 1.4826.
Proof:
Let m be the median of a distribution. Then P(x ≤ m) = .5.
Since the median and mean are the same for a normal distribution N(μ, σ), we have MAD = median {|x–m|} = median {|x–μ|}, and so
where Φ(z) = NORM.S.DIST(z,TRUE).
Since
it follows that
and so
Thus
σ = k ⋅ MAD
Reference
Wikipedia (2024) Median absolute deviation
https://en.wikipedia.org/wiki/Median_absolute_deviation#Relation_to_standard_deviation