I know the mean of a data array, and I know the standard deviation. It is normal distribution. Help?

Question

I need a formula to find probabilities based off a number I am given that falls somewhere in the distribution. Say the mean is 20 and stdev is 2, I need to find the probability of 5 happening. Also need to find it vice versa- I need to know what the probability that a randomly chosen number will be higher than say 8. Thanks for the help

Answer 1

Let m be the mean, s the standard deviation and x your number.

Define z= (x-m)/s, in your case z=(5-20)/2 = 7.5

There exist tables to convert these z-values to percentages, like here:

http://www.isixsigma.com/library/content...

I'll work you through a better example, because the normal distribution is continuous, meaning that one can ask only for the probability of finding x in a certain interval like x>25, or 15<x<22.  

So, suppose m=20, s=2 Probability for x< 23  ?

z= (23-20)/2 =1.5

In the table we find the value 0.4332 for z=1.50 This number means that there is 43.32% probability to find  0<z<1.5 or m<x<m+1.5*s .

We wanted x<23, therefore we half to add 50%, which is the probability for x to be smaller than the mean m.

I hope this helped.