F04
How to find Normal probabilities and quantiles using Table B.3.
Given µ and σ 2 , and that X ~ N ( µ , σ 2 ) , to find P (a ≤ X ≤ b) for given numbers a and b.
1.
Reexpress the probability in terms of the Standard Normal random variable Z ~ N (0, 1) . The
applicable change of variables is
Z = ( X − µ) /σ
and the result is
P ( a ≤ X ≤ b) = P ( a − µ ≤ X − µ ≤ b − µ )
X −µ
b−µ
a−µ
=P
≤
≤

σ
σ 
 σ
b−µ
a−µ
= P
≤ Z ≤

σ 
 σ
2.
a−µ

To find the probability in Step 1 extract two values from the Table B.3, P  Z ≤

σ 

b−µ

and P  Z ≤
.
σ 

Then the desired probability is found as
b−µ
a−µ


P ( a ≤ X ≤ b) = P  Z ≤
 − P Z ≤
.
σ 
σ 


Example:
X ~ N (2, 9) . Find P (1 ≤ X ≤ 2) .
2−2 
 1− 2
P (1 ≤ X ≤ 2 ) = P 
≤ Z ≤

3 
 3
= P ( − 1/ 3 ≤ Z ≤ 0 )
= P ( Z ≤ 0 ) − P ( Z ≤ − 1/ 3 )
= 1 / 2 − 0.3707
= 0.1293
≈ 0.13
Finding Normal probabilities is easy, just follow the two-step procedure.
Now consider the inverse problem, namely given a probability in a Normal distribution (a p-value),
find the corresponding quantile in the Normal population. I.e., given probability p find x such that
P ( X ≤ x) = p
Again the procedure is two-step.
1.
x−µ

Reexpress in terms of Z. P ( X ≤ x) = P  Z ≤
 ≡ P ( Z ≤ z) = p
σ 

2.
For the given p, go to the Table B.3 to find z. Solve z = ( x − µ ) / σ for x.