I'm new to studying z-scores and I've been told that for a gaussian statistic, around 95% of the val

Callum Dudley 2022-07-04 Answered
I'm new to studying z-scores and I've been told that for a gaussian statistic, around 95% of the values lie within the area two standard deviations above and below the mean, which (in accordance to my interpretation) would imply,
μ 2 σ μ + 2 σ A e ( ( x μ ) / σ ) 2 d x = 0.95 + A e ( ( x μ ) / σ ) 2 d x
Firstly, am I correct in my presumption? and secondly, is there any way to calculate the integral on the left to prove this point mathematically?
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Answers (1)

Caiden Barrett
Answered 2022-07-05 Author has 20 answers
Using a change of variables x = μ + σ t, and inserting the missing 1/2 in the exponentials, the left side can be evaluated using the error function
A 2 2 e t 2 / 2 d t = 2 π A erf ( 2 )
while the right side is 2 π A. In fact erf ( 2 ) 0.9544997360.
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