Percent of Data Values in 215-305 Range | Chebyshev's Theorem

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The discussion focuses on calculating the percentage of data values within the range of 215 to 305 for a distribution with a mean of 260 and a standard deviation of 18. It emphasizes that the assumption of a normal distribution is crucial for accurate calculations. The analysis shows that at least 20% of the distribution lies between the endpoints of the specified range, based on weights assigned to the mean and endpoints. The conversation also references Chebyshev's theorem to validate the findings. The conclusion highlights the importance of understanding the distribution's characteristics when applying statistical theorems.
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a distribution has a mean of 260 and a standard deviation of 18 What is the percentage of data values that will fall in the range of 215 to 305 please simple or explain
 
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I'm going to answer your question but I'm making no assumptions while your teacher probably wanted you to assume a normal distribution.

So 260 is exactly in the middle of the interval. Parts of the distribution outside the interval will have the least effect on the standard deviation if they are at the end points. Similarly points inside the distribution will have the least effect if they are at the mean.

Therefore, Assume the distribution has either the weight at the endpoints or at the mean. Let w_m be the weight at the mean and w_e be the weight at an end points. Then w_m+2*w_e=1.

Also

2* \left({305-215 \over 2} \right)^2w_e=18

45w_e=18 <=> w_e=0.4

and therefore:

w_m=1-2*0.4=0.2

Therefore at least 20% of the distribution lies between the endpoints.
 

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