Bunuel wrote:
Competition Mode Question
If set X and set Y consist of more than one element, what is the ratio of the standard deviation of set X to the standard deviation of set Y?
(1) Set X consists of consecutive multiples of 5 and set Y consists of consecutive multiples of 3
(2) The number of elements in sets X and Y are equal
Are You Up For the Challenge: 700 Level QuestionsM36-138
Official Solution:If set X and set Y consist of more than one element, what is the ratio of the standard deviation of set X to the standard deviation of set Y? (1) Set X consists of consecutive multiples of 5 and set Y consists of consecutive multiples of 3
If set X is say {5, 10} and set Y is say {3, 6, 9, ..., 300}, then the standard deviation of set X will be less than that of set Y (because variation of numbers from the mean in set X is less than the variation of numbers from the mean in set Y) and thus the ratio would be less than 1. But if set X is say {5, 10, ..., 500} and set Y is say {3, 6}, then the standard deviation of set X will be more than that of set Y and thus the ratio would be more than 1. Not sufficient.
(2) The number of elements in sets X and Y are equal
Clearly insufficient. We don't know anything about numbers in the sets.
(1)+(2) Two important properties:
If we multiply (or divide) all the numbers in a set by a non-zero constant, the standard deviation will be respectively multiplied or divided by the absolute value of that constant. For example, if the SD of \(\{a, \ b, \ c\}\) is \(s\), then the SD of \(\{2a, \ 2b, \ 2c\}\) will be \(2s\) and the SD of \(\{-3a, \ -3b, \ -3c\}\) will be \(s*|-3|=3s\).
If we add (or subtract) a constant to all the numbers in a set, the standard deviation will NOT change. For example, if the SD of \(\{a, \ b, \ c\}\) is \(s\), then the SD of \(\{a+4, \ b+4, \ c+4\}\) or of \(\{a-1, \ b-1, \ c-1\}\) will also be \(s\).
From the second property it follows, that any \(n\) consecutive multiples of \(x\) will have the same standard deviation. For example, the SD of \(\{3, \ 6, \ 9\}\) will be the same as the SD of \(\{-12, \ -9, \ -6\}\) because we can add 15 to each number in the second set to get the numbers in the first set.
Back to the question, say there are \(n\) numbers in each set and X = \(\{3, \ 6, \ 9, ..., 3n \}\) and Y = \(\{5, \ 10, \ 15, ..., 5n \}\) (as discussed above, it does not matter which \(n\) multiples of 3 or 5 we consider). Notice that numbers in Y are \(\frac{5}{3}\) times the respective numbers in X. So, according to the first property above, \(SD_A*\frac{5}{3}=SD_B\), thus \(\frac{SD_A}{SD_B}=\frac{5}{3}\). Sufficient.
Answer: C