$$10, 10, 10, 10, 8, 8, 8, 8, 12, 12, 11, y$$The twelve numbers shown represent the ages, in years, the the twelve residences on a details city block. What is the typical age, in years, the the twelve houses on the block?

So, you were trying to be a great test taker and also practice because that the GRE with PowerPrep online. Buuuut climate you had actually some questions about the quant section—specifically concern 12 of ar 4 of exercise Test 1. Those questions experimentation our expertise of Numerical techniques for describing Data deserve to be type of tricky, but never fear, londonchinatown.org has acquired your back!

Survey the Question

Let’s find the trouble for clues regarding what it will certainly be testing, together this will certainly help change our minds come think around what type of math expertise we’ll usage to settle this question. Pay attention to any kind of words that sound math-specific and also anything special about what the number look like, and mark them on your paper.

You are watching: The twelve numbers shown represent the ages

So the difficulty wants to understand the median that a perform of numbers. Median definitely sounds prefer a mathematics term. If we think carefully, we’ll remember the the median is the middle variety of a list of numbers. We must expect the we’ll check our Numerical methods for describing Data math ability solving this question.

What perform We Know?

Let’s closely read through the question and make a perform of the things that us know.

We have actually a perform of $12$ numbersOne of lock is lacking ($y$)We desire to uncover the median of the perform of numbers

Develop a Plan

Let’s an initial refresh our memories about what the typical is. Of course, if we’re emotion confident around finding medians of perform of numbers, we can just skip ideal over this refresher.

Concept Refresher – Median

The mean is the “middle” variety of a perform of numbers. That is the number for which we have the same number of numbers the are greater than or much less than the number in the list. Together an example, the mean of this list of 3 numbers ($1, 2, \and 5$) is $2$, because we have one number greater than $2$ and one number much less than $2$ in that list. There space three procedures we’ll usage for detect the typical of a perform of numbers:

Put the numbers in stimulate from the very least to greatest.Start cross off 2 numbers in ~ a time: very first the the smallest number and also then the largest number. Stop crossing off numbers when we have actually either one or 2 numbers left.If we have only one number left, the mean is that number. If 2 numbers are left, the median is the mean of those 2 numbers.

Median example 1

Let’s begin by detect the mean for the following list of numbers:

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

The very first step is to placed the numbers in bespeak from least to greatest:

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

To find the center number in a list, it makes sense that we would begin cutting off numbers indigenous the 2 ends. Then whatever remains is the middle number! for this reason the second step is to start crossing off numbers 2 at a time: the best number and also the the very least number. So an initial we would cross off the $-7$ and also $17$ from the list.

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

Our list went from 5 numbers down to 3 numbers. Making some progress! However, we still don’t recognize what the center number is. Let’s cross turn off the least and greatest numbers again:

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

Ah ha! only one number is left, therefore that should be the middle number. The average of the perform of numbers ($-7, 3, 8, 14, 17$) is $8$.

Median instance 2

That wasn’t so bad. Let’s uncover the mean of another collection of numbers: $(3, -7, 17, 8, 14, 6)$. First, let’s placed the numbers in stimulate from least to greatest.

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


Alright, climate let’s discover the center number through crossing off the number at the end.

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

Can’t to speak for sure what the middle number is yet, therefore let’s cross turn off the least and greatest numbers again:

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

Only two numbers remaining! Looks together if over there is a difficulty though. If we cross off the greatest and also smallest numbers, climate we’ll have no numbers left! Can’t uncover the middle of an empty perform of numbers. We’ll just need to settle on taking the typical of the two center numbers.

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

Well that’s great to know. If we have actually an even variety of numbers, then us can’t cross the numbers turn off the list two at a time until we have only one number left. Over there will constantly be one even number of numbers left, so we’ll arrangement to take it the typical of the two center numbers. Excellent. Looks together if we’ve arisen a technique for united state to find the median, or the middle number in a perform of numbers. Let’s update our steps for detect the mean to reflect this. To discover the average of a list of numbers:

Put the number in stimulate from the very least to greatest.Start cross off two numbers at a time: an initial the smallest number and also then the largest number. Prevent crossing off numbers when we have either one or 2 numbers left.If we have only one number left, the median is the number. If 2 numbers are left, the median is the mean of those two numbers.Note: If there is one odd variety of numbers, then the mean will be the value of the solitary number in the middle. If over there is one even variety of numbers, then we’ll need to discover the mean of the two numbers in the middle.

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

We desire to uncover the typical of a list of twelve numbers. We know that to find a typical we should very first put the numbers in stimulate from least to greatest. Then for a list v an even number of numbers, we store crossing turn off numbers two at a time (the greatest and also least numbers) until just two numbers space left. Let’s begin by placing the number in order from least to greatest:

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

Um, but where have to we put the $y$? The beginning? Wait, or perhaps the end? Hmmm…we don’t understand it’s value, so we can’t really encompass it once ordering the number from the very least to greatest. If the numbers are put in bespeak from the very least to greatest, the $y$ might be the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, or 12th number. So plenty of possibilities! of course, we can find the median for every twelve possibilities, yet that sounds very time-consuming. Looks as if math-ing the end this problem isn’t the most efficient solution, for this reason let’s try to uncover a logical way to fix this concern instead.

As through all numeric entry questions, us are required to form in a number answer. Due to the fact that we don’t know the value of $y$, it can’t it is in the median. Otherwise us would need to kind in a numerical worth that we don’t know, i m sorry is impossible! Also, the truth that we can form in one actual numerical answer kind of implies that the worth of $y$ doesn’t matter. There isn’t a “can’t be determined” alternative for this question. Thus, us should be able to put $y$ almost everywhere that we want in the list, find the middle number, and report that together our median. Let’s arbitrarily just put $y$ as the very first number, then cross off the least and also greatest numbers until we only have the center two numbers remaining. Then we have the right to average castle to uncover the median!

Solve the Question

Crossing off the least and also greatest numbers, two complete numbers in ~ 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 just two numbers remaining, the median must it is in the average of these 2 numbers. The average of $10$ and $10$ would be…wait because that it…$10$! for this reason the exactly answer is $10$.

What Did us Learn

Medians! no so dreadful to transaction with. Just put the number of a perform in bespeak from the very least to greatest, cross off, in pairs, the largest and also smallest numbers of the list, and we have actually our median. Just need come remember to take it the mean of the two center numbers remaining at the end if our list has actually an even variety of values.

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