The procedure for finding the standard deviation for a set is as follows:
1) Find the difference between each term in the set and the mean of the set.
2) Average the squared "differences."
3) Take the square root of that average.
Notice that the standard deviation hinges on step 1: finding the difference between each term in the set and the mean of the set. Once this is done, the remaining steps are just calculations based on these "differences."
Thus, we can rephrase the question as follows: "What is the difference between each term in the set and the mean of the set?"
(1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:
[ x – 10, x – 9, x – 8, x – 7, x – 6, x – 5, x – 4, x – 3, x – 2, x – 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]
Notice that the difference between the mean ( x) and the first term in the set ( x – 10) is 10. The difference between the mean ( x) and the second term in the set ( x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!
(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:
Sum of the squared differences:
102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 02 + (-1)2 + (-2)2 + (-3)2 + (-4)2 + (-5)2 + (-6)2+ (-7)2 + (-8)2 + (-9)2 + (-10)2 = 770
Average of the sum of the squared differences:
770
21
= 36
2
3
The square root of this average is the standard deviation: ≈ 6.06
(2) NOT SUFFICIENT: Since the set is consecutive, we know that the median is equal to the mean. Thus, we know that the mean is 20. However, we do not know how big the set is so we cannot identify the difference between each term and the mean.
Therefore, the correct answer is A.
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