Data Analysis, Statistics & Probability
Tackling Data Puzzlers
Use this three-step method every time you answer a Data Puzzler item:
Step 1: Identify and arrange the data according to value.
Step 2: Solve for any missing pieces.
Step 3: Give the item what it wants.
Data Puzzler in Slow Motion
Now let’s look at the steps more closely.
7. If average of 13, 6, 9, x, and y is 12, what is the average of x + y?
(A) 6
(B) 9
(C) 12
(D) 16
(E) 32
Step 1: Identify and arrange the data according to value.
We have a set of five numbers, two of which are variables. So let’s go ahead and arrange what we can in order according to value. Because we don’t know what the variables represent, we’ll place them last:
{6, 9, 13, x, y}
This ordering probably won’t help us too much on an average item, but it’s better to be safe than sorry. It would be disastrous to be in the middle of solving an item, only to realize you have to go back to the beginning and start over.
Step 2: Solve for any missing pieces.
Remember the mean/average equation: . On all items involving averages, you need to solve for all three pieces of the equation before moving on to the solution. Any two pieces supplied in the stem will solve for the third. Referring back to the stem, we have five numbers, the average of which is 12. Go ahead and solve for their sum:
Step 3: Give the item what it wants.
Because the item is asking for the average, we must find the sum of x and y and the number of items being averaged. The second part is pretty easy: x and y are being averaged, so the number of items is 2. To find their sum, we need to add together what we know from the set.
6 + 9 + 13 = 28.
Now if three of the numbers in the set add up to 28 and the total sum of the set is 60, then the remaining numbers, x and y, must add up to 60 -– 28, or 32. We’re not quite done yet. 32 looks pretty good, and it’s one of the answer choices, but it’s not what the item asked for. To find their average, divide by 2:
The correct answer is D, 16.
Guided Practice
Try this next one on your own.
Set B = {5, 3, x, 5, 11, 8, 9}
10. If the median of set B is also its average, then what is one possible value of x?
(A) 5
(B) 8
(C) 11
(D) 15
(E) 21
Step 1: Identify and arrange the data according to value.
Are the numbers in this set in no particular order? You know what to do.
Step 2: Solve for any missing pieces.
Is mean/average involved in the item? If so, find all three parts to the equation.
Step 3: Give the item what it wants.
Make sure you have the number the item asked for and not a distractor.
Guided Practice: Explanation
Step 1: Identify and arrange the data according to value.
There are seven numbers all jumbled together in set B, so let’s put those in order.
{3, 5, 5, 8, 9, 11, x}
Because x is unknown, we went ahead and placed it after the 11. This does not mean that x is greater than 11. Realistically, x could rightfully be placed anywhere within the set.
Step 2: Solve for any missing pieces.
Here is where the item becomes a little tricky. We know the average has to equal the median, so let’s take a look at what the possible medians of set B could be. If x is removed from the set, then the two possible medians are 5 and 8.
Let’s start with 5. If 5 is the median, then 5 must also be the average. If 5 is the average, then the sum of all 7 numbers in the set must be or 35. If the sum of all seven numbers is 35, then the solution for x would look like this:
Let’s look at 8. If 8 is the median, then 8 must also be the average. If 8 is the average, then the sum of all 7 numbers in the set must be , or 56. If the sum of all seven numbers is 56, then the solution for x would look like this:
Step 3: Give the item what it wants.
We have discovered two possible values for x. Let’s see if any of them match up with the answer choices. Bingo! Answer choice D is 15. Don’t worry about the –6; the item asked for one possible answer choice. Choose D and move on.
Independent Practice
Work out this item on your own. Then turn the page for an explanation.
14. The average test score of six students is 96. What would a seventh student need to score to reduce the average to 89?
(A) 35
(B) 47
(C) 55
(D) 89
(E) 95
Independent Practice: Explanation
Step 1: Identify and arrange the data according to value.
The data for this item are the unknown test scores of six students. Because there is no way of determining what they are, it is impossible to arrange them in any order. Let’s move on to step 2.
Step 2: Solve for any missing pieces.
We do know that the item involves averages, so let’s look for the pieces to the average formula and solve for any missing information. Because 96 is the average of six students’ test scores, we can solve for the sum of all their test scores:
The second part of the stem informs us that an additional student takes the test—increasing the number of test-takers to seven—and changes the average to 89. So the sum of all seven test-takers must now equal:
Step 3: Give the item what it wants.
The item asks for the test score of the seventh test-taker. If the sum of the previous six test-takers is 576 and the sum of the previous six plus the additional seventh is 623, then the additional test-taker’s score would have to be the difference between 623 and 576. 623 – 576 = 47, answer B.
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