Associated Topics || Dr. Math Home || Search Dr. Math

### IQ Scores by Category

```
Date: 7/28/96 at 9:26:1
From: MIGUEL SPRUILL
Subject: measures of dispersion, the normal distribution

On standard IQ tests, the mean is 100, with standard deviation of 15.
The results come very close to fitting a normal curve. Suppose an IQ
test is given to a very large group of people. Find the percent of
people whose IQ scores fall into the following categories:

1) greater than 115.
2) greater than 145.

If you find both kinds of standard deviation, the sample standard
deviation and the population standard deviation, which of the two will
be a larger number for a given set of data?

Find (a) the range, and (b )the standard deviation for each sample
round fractional answers to the nearest hundredth.

1) 67, 83, 55, 68, 77, 63, 84, 72, 65
```

```
Date: 7/28/96 at 16:6:13
From: Doctor Anthony
Subject: Re: measures of dispersion, the normal distribution

To find the standardised values for use with normal tables you
calculate z from  z = (x-m)/s where m = mean and s = standard
deviation.

For greater than 115 we have

z = (115.5 - 100)/15  = 15.5/15 = 1.0333.

The tables give an area 0.8493, so

tail area = 0.1507.

So 15.07 percent of the population has an IQ above 115.

For greater than 145 we have

z = (145.5 - 100)/15 = 45.5/15 = 3.0333.

The tables give an area .99878, so

tail area = 0.00122.

So 0.12 percent of the population has an IQ above 145.

If you use a sample to estimate population mean and s.d. you lose one
'degree of freedom' in that with mean found, then you no longer have n
independent values for finding the standard deviation.  So in
calculating variance from

SIGMA(x-m)^2/n (sample variance),

you would use

SIGMA(x-m)^2/(n-1) if m and s are calculated from the sample as
estimates for the population.

So the standard deviation is greater if you use the sample to estimate
population standard deviation.

Find range and standard deviation of the sample 67, 83, 55, 68, 77,
63, 84, 72, 65.

The range = 84 - 55 = 29.

Variance of the sample is calculated from

SIGMA[x^2/n] - mean^2
mean = 70.44
standard deviation = 8.995.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Statistics

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search