In the following example, why would Chebyshev's Theorem be used instead of the Empirical Rule?
The Empirical Rule is a rule in statistics that says for a normal distribution, most of all of the data will land between three standardized yet different deviations from their mean. What the empirical rule does is it displays that 68% of the information will fall inside the first standard deviation, that about 95% of the data will fall within the first two standard deviations, and that 99.7% of the data will fall within the first three standard deviations of the mean. It is also sometimes called the rule of 68-95-99.7 of the 3 Sigma Rule.This rule very closely relates to the diagram (it displays the rule fairly accurately). Chebyshev's Theorem states essentially that a distribution of any shape or size puts a lower level on the percents of the observations. This occurs inside a provided number of standard deviations which come from its mean. Now the Empirical Rule on the other hand applies to more specifically mounded-shaped and or symmetrical distributions. It displays percentages as approximations inside of a standard deviation which comes from its mean. The Emperical Rule could apply to the gas mileage on Subaru Imprezzas and Chebyshev's Theorem could be used for something like trying to calculate what percentages of a value will end up between 159 and 227 for a data set with mean of 194 and standard deviation of 16.5.© BrainMass Inc. brainmass.com October 2, 2022, 6:57 am ad1c9bdddf
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The reason that the Chebyshev's Theorem would be used instad of the Empirical Rule is that Chebyshev's Theorem is valid for any set of data.
Where any set of data within the K standard deviations of the mean is demonstrated 1-1/K², K being any number greater than 1.
Example: When K = 2 the formula will demonstrate that 75% of the data will always be within two standard deviations of the mean.
When K = 3 the formula will demonstrate that 89% of the data must always be within three standard deviations of the mean.
The Empirical Rule is used only for data that have a normal distribution. Its property facts include:
A standard normal distribution that demonstrates (also know as bell-shaped):
"68.2% of the data lies within 1 standard deviation of the means.
95.4% of the data lies within 2 standard deviation of the means.
99.7 of the data lies within 3 standard deviation of the means."
Also see : (http://www.andrews.edu/~calkins/math/edrm611/edrm04.htm)
I don't have a visual of the diagram, but if it is a scatter diagram and is displaying a distorted curve all you can go by is Chebyshev's Theorem. As you can see above there is quite a bit of difference of percentages within the standard deviations of the means between Chebyshev's Theorem and the Empiracle Rule.
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