Find Mean and Standard Deviation from 2 given percentile and values

Question

Consider a normal distribution curve where the 65th percentile is at 12 and the 45th percentile is at 9, how do I find the mean and standard deviation

Answer 1

You can get the two z values from the percentiles using the normal inverse function.

For example, in Excel, z₁ =NORM.S.INV(0.45) = -0.1257 and

z₂ =NORM.S.INV(0.65) = 0.3853

Then, recall z₁ = (9 - µ)/σ and z₂ = (12 - µ)/σ

Thus, z₂ - z₁ = 3/σ

Yet, z₂ - z₁ = 0.3853 - -0.1257 = .5110

Thus, 3/σ = .5110 , so σ = 3/.5110 = 5.8711

Then, as z₂ = (12 - µ)/σ = 0.3853, µ = 12 - 0.3853 σ = 12 - 0.3853 * 5.8711 = 9.7378