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

### Standard Deviation of Sample

```
Date: 01/12/98 at 17:37:49
From: Howard Amerlan
Subject: Standard deviation of sample

I have read your proof of how the standard deviation of a sample is
found, and I have a question about how to describe what is happening
to students.  For example, if some height is normally distributed with
a mean of 69 in. and a standard deviation of 5.5 in. (this is the
population information), then suppose we take a simple random sample
of 100 from this distribution. Then supposedly the mean should still
be 69 but the standard distribution of the sample will be 0.55
(5.5 / sqrt 100). How do we describe to the student that this sample
has a narrower variance, hence a narrower standard deviation, than the
entire population has?  This is especially confusing since the law of
large numbers says that the larger the number of samples, the closer
we get to the true mean and standard deviations.
```

```
Date: 01/12/98 at 18:53:26
From: Doctor Anthony
Subject: Re: Standard deviation of sample

The expression 5.5/sqrt(100) is the s.d of the MEANS of samples size
100.

This is quite different from the s.d of individual measurements of
heights from the sample. We are dealing with a completely different
variable made up of the MEANS of samples size 100. It is commonly
referred to as the 'standard error of the mean'. If you imagine
collecting say 20 samples of size 100 and working out the mean of each
sample, then these MEANS will form a new population known as the
sampling distribution and will be distributed with a much smaller s.d.
than will the heights of individual persons. For samples of size 100
the s.d. of the sampling distribution will be 1/10 that of individual
people.

-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

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/