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I was reading "We Don’t Quite Know What We Are Talking About When We Talk About Volatility" by Goldstein and Taleb, and I was trying to quickly verify numerically the relation between mean deviation and standard deviation.

However, I get that 0.8 is the ratio between mean deviation and variance, not mean deviation and standard deviation. See code example below.

Can anybody explain to me what I am doing wrong?

import numpy as np

n = 10000
x = np.random.normal(0, 1, size=[n, 1])
sum(abs(x)) / sum(x ** 2)  # approx 0.8
sum(abs(x)) / sum(x ** 2) ** 0.5  # approx 80

enter image description here

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1 Answer 1

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There's a small typo,

Mean absolute deviation, with 0 mean = sum(abs(x))/n

Standard deviation, with 0 mean = np.sqrt(sum(x ** 2))/np.sqrt(n)

So when you divide MAD over SD you should use:

sum(abs(x)) / (np.sqrt(n) * np.sqrt(sum(x ** 2)))

which gives 0.8 as expected

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