So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 12 of Section 4 of Practice Test 1. Those questions testing our knowledge of Numerical Methods for Describing Data can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on your paper.

So the problem wants to know the median of a list of numbers. Median definitely sounds like a math term. If we think carefully, we’ll remember that the median is the middle number of a list of numbers. We should expect that we’ll test our Numerical Methods for Describing Data math skill solving this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

1. We have a list of $12$ numbers
2. One of them is missing ($y$)
3. We want to find the median of the list of numbers

Develop a Plan

Let’s first refresh our memories about what the median is. Of course, if we’re feeling confident about finding medians of lists of numbers, we can just skip right over this refresher.

Concept Refresher – Median

The median is the “middle” number of a list of numbers. It is the number for which we have the same number of numbers that are greater than or less than that number in the list. As an example, the median of this list of three numbers ($1, 2, \and 5$) is $2$, because we have one number greater than $2$ and one number less than $2$ in that list. There are three steps we’ll use for finding the median of a list of numbers:

1. Put the numbers in order from least to greatest.
2. Start crossing off two numbers at a time: First the smallest number and then the largest number. Stop crossing off numbers when we have either one or two numbers left.
3. If we have only one number left, the median is that number. If two numbers are left, the median is the average of those two numbers.

Median Example 1

Let’s start by finding the median for the following list of numbers:

$$3, -7, 17, 8, 14$$

The first step is to put the numbers in order from least to greatest:

$$-7, 3, 8, 14, 17$$

To find the middle number in a list, it makes sense that we would start cutting off numbers from the two ends. Then whatever remains is the middle number! So the second step is to begin crossing off numbers two at a time: the greatest number and the least number. So first we would cross off the $-7$ and $17$ from the list.

$$-7, 3, 8, 14, 17$$
$$3, 8, 14$$

Our list went from five numbers down to three numbers. Making some progress! However, we still don’t know what the middle number is. Let’s cross off the least and greatest numbers again:

$$3, 8, 14$$
$$8$$

Ah ha! Only one number is left, so that must be the middle number. The median of the list of numbers ($-7, 3, 8, 14, 17$) is $8$.

Median Example 2

That wasn’t so bad. Let’s find the median of another set of numbers: $(3, -7, 17, 8, 14, 6)$. First, let’s put the numbers in order from least to greatest.

$$-7, 3, 6, 8, 14, 17$$

Alright, then let’s find the middle number by crossing off the numbers at the end.

$$-7, 3, 6, 8, 14, 17$$
$$3, 6, 8, 14$$

Can’t say for sure what the middle number is yet, so let’s cross off the least and greatest numbers again:

$$3, 6, 8, 14$$
$$6, 8$$

Only two numbers remaining! Looks as if there is a problem though. If we cross off the greatest and smallest numbers, then we’ll have no numbers left! Can’t find the middle of an empty list of numbers. We’ll just have to settle on taking the average of the two middle numbers.

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Median= \Average \of 6 \and 8$$
$$\;\;\;\;\;\;\Median = {6+8}/2$$
$$\;\;\Median = {14}/2$$
$$\Median = 7$$

Well that’s good to know. If we have an even number of numbers, then we can’t cross the numbers off the list two at a time until we have only one number left. There will always be an even number of numbers left, so we’ll plan to take the average of the two middle numbers. Excellent. Looks as if we’ve developed a method for us to find the median, or the middle number in a list of numbers. Let’s update our steps for finding the median to reflect this. To find the median of a list of numbers:

1. Put the numbers in order from least to greatest.
2. Start crossing off two numbers at a time: First the smallest number and then the largest number. Stop crossing off numbers when we have either one or two numbers left.
3. If we have only one number left, the median is that number. If two numbers are left, the median is the average of those two numbers.
4. Note: If there is an odd number of numbers, then the median will be the value of the single number in the middle. If there is an even number of numbers, then we’ll need to find the average of the two numbers in the middle.

Now that we’ve refreshed our memory about medians, let’s get back to the problem!

We want to find the median of a list of twelve numbers. We know that to find a median we should first put the numbers in order from least to greatest. Then for a list with an even number of numbers, we keep crossing off numbers two at a time (the greatest and least numbers) until only two numbers are left. Let’s start by putting the numbers in order from least to greatest:

$$8, 8, 8, 8, 10, 10, 10, 10, 11, 12, 12$$

Um, but where should we put the $y$? The beginning? Wait, or maybe the end? Hmmm…we don’t know it’s value, so we can’t really include it when ordering the numbers from least to greatest. If the numbers are put in order from least to greatest, the $y$ could be the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, or 12th number. So many possibilities! Of course, we could find the median for all twelve possibilities, but that sounds very time-consuming. Looks as if math-ing out this problem isn’t the most efficient solution, so let’s try to find a logical way to solve this question instead.

As with all numeric entry questions, we are required to type in a numerical answer. Since we don’t know the value of $y$, it can’t be the median. Otherwise we would need to type in a numerical value that we don’t know, which is impossible! Also, the fact that we can type in an actual numerical answer kind of implies that the value of $y$ doesn’t matter. There isn’t a “can’t be determined” option for this question. Thus, we should be able to put $y$ anywhere that we want in the list, find the middle number, and report that as our median. Let’s arbitrarily just put $y$ as the first number, then cross off the least and greatest numbers until we only have the middle two numbers remaining. Then we can average them to find the median!

Solve the Question

Crossing off the least and greatest numbers, two total numbers at a time, we get:

$$y, 8, 8, 8, 8, 10, 10, 10, 10, 11, 12, 12$$
$$8, 8, 8, 8, 10, 10, 10, 10, 11, 12$$
$$8, 8, 8, 10, 10, 10, 10, 11$$
$$8, 8, 10, 10, 10, 10$$
$$8, 10, 10, 10$$
$$10, 10$$

With only two numbers remaining, the median must be the average of these two numbers. The average of $10$ and $10$ would be…wait for it…$10$! So the correct answer is $10$.

What Did We Learn

Medians! Not so awful to deal with. Just put the numbers of a list in order from least to greatest, cross off, in pairs, the largest and smallest numbers of the list, and we have our median. Just need to remember to take the average of the two middle numbers remaining at the end if our list has an even number of values.

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