Area of a Normal Distribution

μ

15

σ

2

z

-1

large format

To find

P(X≤x)

for a normally distributed random variable

X

with mean

μ

and standard deviation

σ

we standardize values from the

N(μ,σ)

distribution using

z=

x-μ

σ

so that

P(X≤x)=P(Z≤z)

where

Z

is a standard normal random variable. Such probabilities are represented as areas to the left of

x

or

z

under a corresponding density curve.This Demonstration provides a visualization of the relationship between a normal distribution

N(μ,σ)

and the standard normal distribution

N(0,1)

. Specifically, the area to the left of the value

x

in a

N(μ,σ)

distribution corresponds to an area to the left of the value's

z

–score in a standard normal

N(0,1)

distribution. Likewise, the areas to the right of

x

and

z

correspond to

P(X≥x)=P(Z≥z)

.