CDF from Z score

z score:      CDF=    % of data less than this Z score (= area under the standard normal curve from -∞ to this Z score)

Calculate Z-score.

z = (x - x̄) / s

x:      μ x̄:      σ s:

Quartiles and deciles of the standard normal distribution.

All normal curves: area under the curve equals 1.

The standard normal distribution: mean μ=0. standard deviation σ=1.

Any Z score is the number of standard deviations from the mean 0 of this standard normal curve..

Function of x:

(vertical exaggerated to see the "bell"):

f(0)=1/√(2π) ≈.3989
√(2π)≈2.507     √π≈1.772

y=1/(sqrt(2*PI)) * e^(-x^2/2)  
Alternate definitions of what's standard.

example normal distributions

Function of x. μ and σ determine each particular curve.

μ=100, blue σ=10, red σ=5

y=1/(10*sqrt(2*PI)) * e^(-(x-100)^2/(2*10^2))  ; y=1/(5*sqrt(2*PI)) * e^(-(x-100)^2/(2*5^2))  

Xmin: 80  Xmax:120   Ymin:-.01   Ymax: .1 

u=10, blue s=1, red s=2, green s=3
y=1/(1*sqrt(2*PI)) * e^(-(x-10)^2/(2*1^2))  ; y=1/(2*sqrt(2*PI)) * e^(-(x-10)^2/(2*2^2))  ; y=1/(3*sqrt(2*PI)) * e^(-(x-10)^2/(2*3^2))  
Xmin: 0  Xmax:20   Ymin:-.1   Ymax: .4