## List M (not shown) consists of 8 different…

## Solution:

We are given a list of numbers: 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and are told that list M (not shown) consists of 8 of these 10 integers. We must determine the standard deviation of the numbers in list M. If we can determine the exact values in list M, we can determine the standard deviation of list M.

**Statement One Alone: **

The average (arithmetic mean) of the numbers in list M is equal to the average of the numbers in the list shown.

We are given the following values for the list shown:

4, 6, 8, 10, 12, 14, 16, 18, 20, 22

We see that the above list is an evenly spaced set. Thus, we can determine the average of the list by using the following equation:

average = (smallest integer in the set + largest integer in the set)/2

average = (4 + 22)/2

average = 26/2 = 13

Thus, the average of list M is also 13. Because list M has 2 fewer numbers than the given set of 10 integers, the sum of the numbers in list M must be 2 x 13 = 26 less than the sum of the numbers in the given set. In other words, the two missing numbers must add up to 26. Well, we know the following pairs of numbers in the given set add up to 26: 4 and 22, 6 and 20, 8 and 18, etc. However, because we do not know which pair is actually missing, we don’t know the actual values in list M. For example, if list M is missing 4 and 22, the list M consists of 6, 8, 10, 12, 14, 16, 18 and 20. If list M is missing 6 and 20, the list M consists of 4, 8, 10, 12, 14, 16, 18 and 22. Since we don’t know the actual values in list M, we cannot determine the standard deviation. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

In general, it’s good to know that two data sets can have the same average, but still have different standard deviations. For example, the set 5, 10, 15, 20, 25 and the set 13, 14, 15, 16, 17 both have the same average (15), but the standard deviations are not the same because the distances of the numbers in the list from the average are different. Notice that the differences of the numbers from the average in the first set are:

-10, -5, 0, 5, 10, whereas those differences in the second set are: -2, -1, 0, 1, 2.

**Statement Two Alone: **

List M does not contain 22.

Although we now know that list M does not contain 22, we still do not have enough information to determine the exact values in list M because list M can contain any 8 of the remaining 9 values in the list shown: 4, 6, 8, 10, 12, 14, 16, 18, 20.

Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

**Statements One and Two Together: **

Using the information from statements one and two, we know that list M has an average of 13 and that it does not contain 22 from the shown list. Because we know that 22 is not in list M, we know that 4 is also not in list M because 4 + 22 = 26 (recall that in statement one we say the two missing numbers must add up to 26). We now know that list M must consist of: 6, 8, 10, 12, 14, 16, 18, 20.

Since we have all the values of list M, we have enough information to determine the standard deviation of list M.

**Answer:** C