Percentage Less than a Given Value

Date: 11/22/97 at 00:30:29
From: Joseph Green
Subject: Basic descriptive statistics

Dear Dr. Math,

If some data are normally distributed, and I know the mean and the 
standard deviation, how do I find out what percentage of the sample is 
less than (or greater than) a given value?

For example, if I'm told that the mean is 12 and the standard 
deviation is 9, how do I find out what percentage of the sample is 
less than zero (or greater than 14, etc.)?

I thought a t table might be useful for this, but even when I use the 
row with infinite degrees of freedom (equivalent to z?) I can't figure 
out how to get an exact answer.

(Can't help thinking that I'm overlooking something very simple here.)

I'd appreciate any help you can give.

Sincerely,
Joseph Green

Date: 11/22/97 at 06:20:49
From: Doctor Mitteldorf
Subject: Re: Basic descriptive statistics

Dear Joseph,

The answer you're looking for comes from an integral of the normal 
function  y = exp(-x^2/2).  The integral is called the error function, 
erf, and is tabulated in any book of statistics or mathematical 
tables. But there's no "formula" for the integral.

For example, if the mean is 12 and the sd is 9 and you want to find 
out what percentage of your sample is less than 0, you first calculate 
that 0 is 1.333 sd's out from the mean, and look up 1.333 in your 
table.  In the table I'm looking at, it says that .4088 is the area 
between the mean and 1.333 sd's, and I know that the area on each side 
of the center is exactly .5, so that leaves .0912 as the proportion 
that is more than 1.333 sd's to the left of the mean.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/