Relationship between STD and MAD

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(xm) = .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

STD- MAD part 1

STD-MAD part 2

STD-MAD part 3

where Φ(z) = NORM.S.DIST(z,TRUE).

Since

STD-MAD part 4

it follows that

STD-MAD part 5

and so

STD-MAD part 6

Thus

σ = k ⋅ MAD

Reference

Wikipedia (2024) Median absolute deviation
https://en.wikipedia.org/wiki/Median_absolute_deviation#Relation_to_standard_deviation

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