Help understanding the Empirical Rule & Chebyshev Theory

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I'm having trouble distinguishing the similarities and the differences between the Empirical Rule and Chebyshev’s Theory. I'm a long time lurker here and figured this would be the place to ask. I understand that Chebyshev's Theory deals with real world distributions and Empirical Rule deals with normal distributions but I can't really distinguish what else to say about it... Please help thanks!
 

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  • #2
statdad
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You essentially nailed it. The Empirical Rule is simple a condensed set of 'rules' (guidelines would be a better term') about the approximate percentages that are found with 1, 2, and 3 standard deviations of the mean for a normal distribution. It is not a mathematical theorem.

Chebyshev's Theorem, on the other hand, IS a theorem - there is a proof of the result: the only requirement is that the distribution have a finite variance. Chebyshev's theorem holds for any distribution, symmetric or skewed. It's most important use (IMO) is not in data description but in more theoretical settings.
 
  • #3
I just can't grasp how to figure out proportions of measurements below a certain number. For instance: Set data has mean of 75 and standard deviation of 5. No info about size of data set or shape of distribution (therefore use chebyshev's).
1. What can you say about proportions of measurements between 60 and 90. (I got 89%).
2. Between 65 and 85. ( I got 75%)
3. Above 90? This is where I get stuck! Can someone please help me?
 
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  • #4
statdad
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I just can't grasp how to figure out proportions of measurements below a certain number. For instance: Set data has mean of 75 and standard deviation of 5. No info about size of data set or shape of distribution (therefore use chebyshev's).
1. What can you say about proportions of measurements between 60 and 90. (I got 89%).
2. Between 65 and 85. ( I got 75%)
The only improvement I would make on these numbers is to say at least [itex] 89\% [/itex] and at least [itex] 75\%[/itex] - Chebyschev's Theorem gives a lower bound on the trapped percentages.
3. Above 90? This is where I get stuck! Can someone please help me?

Remember from part (1) of your question that at least [itex] 89\% [/itex] of the scores are between 65 and 90. Since you can't assume anything about the shape of the distribution, the best you can say is this: we're still missing a maximum of [itex] 11\% [/itex]
of the data. It's possible that all of it missing data is above 90, so the only conclusion to make is at most [itex] 11\%[/itex] of the scores are above 90
 
  • #5
ah ha. Well that seems almost to easy. Thanks statdad, I really appreciate the help!
 

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