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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 

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 

-Doctor Anthony,  The Math Forum
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Associated Topics:
High School Statistics

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