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timer Statistics
A researcher computed the mean, the median, and the standard deviation for a set of test scores. If 50 were to be added to each score, which of these three statistics would change?
A The mean only B The median only C The standard deviation only D The mean and the median E The mean and the standard deviation
A researcher computed the mean, the median, and the standard deviatio
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21 Jul 2021, 06:22
Expert Reply
Top Contributor
Theory
Mean: If all the numbers in the set are increased/decreased by the same number(k) then the mean also gets increased/decreased by the same number(k)
Ex: Suppose the set is {a,b,c,d,e} then the Mean = \(\frac{(a+b+c+d+e)}{5}\) Now, lets increase all the numbers by 50. So, the new set is {a+50,b+50,c+50,d+50,e+50) New Mean = \(\frac{(a+50 +b+50 +c+50 +d+50 + e+50)}{5}\) = \(\frac{(a+b+c+d+e + 5*50)}{5}\) = \(\frac{(a+b+c+d+e)}{5}\) + 50 = Old Mean + 50
Median: If all the numbers in the set are increased/decreased by the same number(k) then the median also gets increased/decreased by the same number(k)
Ex: Suppose the set is {a,b,c,d,e} (in ascending order) then the Median= middle term = c Now, lets increase all the numbers by 50. So, the new set is {a+50,b+50,c+50,d+50,e+50) New Median= middle term = c+50 = Old Median + 50
Standard Deviation: If all the numbers in the set are increased/decreased by the same number(k) then the Standard Deviation DOES NOT CHANGE!
Ex: Suppose the set is {a,b,c,d,e} then the Mean = \(\frac{(a+b+c+d+e)}{5}\) = M SD = \(\sqrt{\frac{(M-a)^2 +(M-b)^2+(M-c)^2+(M-d)^2+(M-e)^2}{5}}\)
Now, lets increase all the numbers by 50. So, the new set is {a+50,b+50,c+50,d+50,e+50) New Mean = \(\frac{(a+50 +b+50 +c+50 +d+50 + e+50)}{5}\) = M+50 New SD = \(\sqrt{\frac{(M+50- (a+50))^2 +(M+50-(b+50))^2+(M+50-(c+50))^2+(M+50-(d+50))^2+(M+50-(e+50))^2}{5}}\) = \(\sqrt{\frac{(M-a)^2 +(M-b)^2+(M-c)^2+(M-d)^2+(M-e)^2}{5}}\) = OLD SD
So, Answer will be D Hope it helps!
Watch the following video to Learn the Basics of Statistics
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