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Normal Distribution v. Student's T Distribution

by kimberley
Tags: distribution, normal, student
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May30-07, 06:46 PM
P: 14
The "Empirical Rule" states that if your data is normally distributed, 95.45% of that data should fall within "2" standard deviations of your Mean. There doesn't appear to be any reference to sample size in the literature regarding the Empirical Rule and a Normal Distribution.

By contrast, however, the Student's T Distribution table, for a two-tailed test, has multipliers that differ from the Empirical Rule. Although where N=10000, at 9999 degrees of freedom, the .0455 level is "2" sd like the Empirical Rule, where N=20, at 19 degrees of freedom, the .0455 level is "2.14" sd.

In sum, then, I don't understand the difference between the "normal distribution" and the "Student's T-Distribution". Is the difference that the Empirical Rule assumes that your data is both normal and "stationary" whereas the Student's T Distribution (i.e., degrees of freedom) assumes that your data is not stationary and that your Mean and Standard Deviations for any period of N will shift with the addition of new data? It's the only thing I can think of since the formulas for confidence intervals for Means and prediction intervals for individual outcomes use the numbers from the Student's T-Distribution.

Thanks in advance.

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Student's distribution arises when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data. Textbook problems treating the standard deviation as if it were known are of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.
Clever Hans
Mar28-11, 07:13 PM
P: 1
I believe that the Student distribution does not assume that the sample mean is the true (underlying) mean. So it is not just the variance or SD that is taken from the data, and I would say that the fact that the sample mean is used is more important than that the sample standard deviation is estimated from the data.

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