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### 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/
```
Associated Topics:
High School Statistics

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