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/
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