Questions About Standard Deviation
Date: 9/20/95 at 16:20:1 From: Brad Morris Subject: standard deviations Dr. Math, One of my colleagues here at Friends' Central School in Wynnewood said a question came up in his precalculus class which he could not answer. It's in two parts: 1. Why does a standard deviation take the square root of the average of the sum of the squared differences between X and X mean instead of the average of the absolute value differences? 2. What is the reasoning behind dividing by n vs. n-1 in the population versus sample standard deviations? Well, I couldn't come up with a satisfactory answer, so I'm ASKING DR MATH. Brad Morris Look forward to hearing from you on this as are the students!
Date: 9/26/95 at 7:53:42 From: Doctor Steve Subject: Re: statistics Here's a "short" answer from Professor Grinstead. Don't hesitate to write back to have something explained.--Steve 1. The reason that squared values are used is so that the algebra is easier. For example, the variance (second central moment) is equal to the expected value of the square of the distribution (second non-central moment) minus the square of the mean of the distribution. This would not be true, in general, if the absolute value definition were used. This is not to say that the absolute value definition is without merit. It is quite reasonable for use as a measure of the spread of the distribution. In fact, I have heard of someone who used it in teaching a course in statistics. (I think he used it because he thought it was a more 'natural' way to measure the spread.) 2. The reason that n-1 is used instead of n in the formula for the sample variance is as follows: The sample variance can be thought of as a random variable, i.e. a function which takes on different values for different samples of the same distribution. Its use is as an estimate for the true variance of the distribution. In statistics, one typically does not know the true variance; one uses the sample variance to ESTIMATE the true variance. Since the sample variance is a random variable, it usually has a mean, or average value. One would hope that this average value is close to the actual value that the sample variance is estimating, i.e. close to the true variance. In fact, if the n-1 is used in the defining formula for the sample variance, then it is possible to prove that the average value of the sample variance EQUALS the true variance. If we replace the n-1 by an n, then the average value of the sample variance is ((n-1)/n) times as large as the true variance. A random variable X which is used to estimate a parameter p of a distribution is called an unbiased estimator if the expected value of X equals p. Thus, using the n-1 gives an unbiased estimator of the variance of a distribution. -Doctor Steve, The Geometry Forum
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