Large Standard Deviations

Date: 12/24/97 at 02:58:36
From: Aaron Peet
Subject: Large Standard Deviations

Is there a way to calculate the percent values derived from the 
z-table without using the table, i.e. a formula?

If not, can you tell me the percent area under 0 and 10 standard 
deviations.  I'm guessing it's pretty close to 100, about 
100-10^(-10).

If there is a graphical formula for the curve I could use calculus 
to find the area, but I can't figure out a formula.

I am specifically trying to disprove (in terms of probability) that a 
person could have an IQ of 300. This is about 13+(1/3) deviations, 
which leaves very little area beyond it 'for someone to exist there'.

Thank you.

Date: 01/08/98 at 09:28:43
From: Doctor Bill
Subject: Re: Large Standard Deviations

Aaron,

The "z-score" for a point in a set of data is the number of standard 
deviations away from the mean the point is. To find the z-score of any 
point you must subtract the mean from that point and then divide by 
the standard deviation for the data.  

So, if x is a point in a set of data that has a mean of X and a 
standard deviation of S, then the z-score of x is;  z = (x-X)/S

In general there is something called the "Empirical Rule" which says 
that "about" 68% of all the data points will be within 1 standard 
deviation of the mean, about 95% will be within 2 standard deviations 
of the mean, and about 99.7% of the data points will be within 3 
standard deviations of the mean. So you can see, it is very unlikely 
that any data points lie to the right of 3 standard deviations, let 
alone 13 standard deviation.

To answer your question, the function for the normal curve is 
f(x) = 1/(sqrt(2*pi))*e^(-.5*x^2), where x is the z score for any data 
point. This function will tell you the probability density (how high 
above the x-axis the curve is) at a given z score. As you can see, if 
you plug in x = 13.33 there is essentially no area between the graph 
and the curve. Therefore, it is HIGHLY unlikely that any person has an 
IQ of 300. Remember, this does NOT prove that it is impossible, but 
statistically we can say that if there is anyone with an IQ of 300, he 
or she is VERY VERY special!

To find the probability that anyone has an IQ of 300 or more, you 
would do the calculation 1 - F(z), where z is the z score of 300 and 
F(z) is the integral from -infinity to z of the function f(x), given 
above.

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