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explain chebyshev's theorm and empirical rule

i'm trying to figure out how to solve a problem. the mean is 50 and the st. deviation is 15. the question is,"what % of the data will fall between 50 and 110? I used the chebyshev's formula (1-1/k 2^)(100%), but not sure if the answer is right? please help!

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Does the problem state that the data is normal?

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1 Answer

Well, Chebyshev tells us: If we are k standard deviations away from mean (smaller or bigger than mean), no more than 1/k2of the values will lie be farther than k standard deviations away from mean.

In your problem, 110 = 50 + 4(15), so we are talking about 4 standard deviations from the mean (k = 4). 

Then, no more than 1/42 = 1/16 of our data points can be more than 4 standard deviations away from the mean (50). So the rest (at least 15/16) of the data points are within 4 st. dev. In percents this is 15/16 • 100 = 93.75% 

But note that we just found that at least 93.75% of our data must be 4 st. dev. smaller OR bigger than our mean of 50. Since we just want the bigger values (50 to 110), our answer is half that, or __ 46.875% __

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This problem does not need empirical rule (and you cannot use it unless the data are assumed to be normal), but the empirical rule states that approx. 68% of data are within 1 standard deviation of mean, 95% of data are within 2 standard deviations of mean, and  99.7% of data lie within 3 standard deviations of mean.

In this data set (if it is normal) with a mean of 50 and st. dev. of 15, that means that

  • 68% of data are in the range 50 ± 15, or (35, 65)
  • 95% of data are in the range 50 ± 30, or (20, 80)
  • 99.7% of data are in range 50 ± 45, or (5, 95)