**Mean and Median**

This is an area of math that is oftentimes given short shrift. Many students think, hey, *I know median and mode—those problems are easy.* On the actual test, though, students oftentimes end up trying to solve a mean problem in a far more laborious and time-consuming way than is necessary. Take a crack at the problem below. See if you can solve it in less than 30 seconds. It’s possible!

*20 students from Mrs. Peterson’s English class take her final, and score an average of 75 points. Mr. Chang, who is known for administering far easier tests, has 30 students, and they score an average of 95 points. If we combine the tests from the two classes what is the average score?
*

*(A) **82*

*(B) **85*

*(C) **87*

*(D) **90*

*(E) **Cannot be determined from information given.*

* *

**The Approach
**

One way to do this problem is to find the total score for each class, and then add these two scores together, dividing by the number of students in both class. Just writing that was a mouthful, so to speak. Actually doing the math would take quite a long time. Sure, the new GRE offers a calculator, but what if there was a way to solve the above problem that was even faster than having to crunch out numbers on a calculator?

When combining two sets of numbers we can quickly find the average if we know the following:

- The number of members in each set.
- The mean of each set.

**The Ratio**

Above, we have 20 students who have a mean of 75 and 30 who have a mean of 95. To find the average, we first look at the ratio of members in each set. Here the ratio is 2:3, because we have 20:30 students across the two classes, which reduces to 2:3.

**The Range **

Next, we look at the range of means: 95 – 75 gives us a range of 20. That is, the difference in means is 20. Now, we want to break up 20 into a 3:2 split. Why? Well, imagine we had an equal number of students. The mean would simply be right in the middle of 75 and 95, which is 85. That is, we split up the range—20—evenly. Ten higher than 75 and ten lower than 95. If we have a 3:2 split, we have to split 20 into 12 and 8. Because more students scored the average of 90, the average will be weighted in the direction of 95. So, we find which point is 12 higher than 75 and eight lower than 95. Notice the ratio of 12: 8 is the same as 3:2. So what number is 12 higher than 75? 87. Notice 87 is eight lower than 95. Therefore our answer is (C).

**Don’t Forget the Powers of Approximation**

An even quicker, though less precise, way of approaching this problem is to remember that the average is going to be skewed towards 95. Right off the bat we can eliminate (A) and (B) because they are too low. (E) is also suspect, because we can definitely determine an answer. (E) suggests some form of ambiguity. That leaves us with (C) and (D). Notice that 90 is skewed a little too close to 95. So, therefore, our answer must be 87.

**Takeaway**

Having a good sense of how averages work will save you lots of time on the actual test. The key, though, is to practice the above approach until you become as comfortable as possible.

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Hi Chris,

First of all, I would like to thank you and the entire Magoosh team for such an awesome ollection of study materials and videos. It makes studying more fun. 🙂

I just had a quick question:

If the ration between the two classes Mrs Peterson’ s and Mr Chang is 2:3, why are we dividing the difference between means i.e. 20 into 3:2 split?

Waiting eagerly for your reply.

Thanks.

Glad you are enjoying the material 🙂

The logic here is that there are more students in Mr. Chang’s class, but in a 3:2 ratio. That is why we want to split it up 20 in that manner (making sure to “flip” the direction so that it is weighted towards Mr. Chang’s class).

This is an intuitive method that some people like, and others don’t. I always tell student that the key is getting the correct answer. So the other option is always to just use the good old formula: [30(95) + 20(75)]/(30+20). With a calculator handy on the GRE, I’d actually recommend this method :).

Hope that helps!

Thanks a lot, Chris.

The explanation is lucid.

I love your blogs, they’re so informative! Also, Chris, you’re really good looking 🙂

Well, thank you! Though I do feel a little abashed about the latter part of the comment :).

Hi Chris you are doing awesome job man! Well I am not able to address u for any comments directly, I get replies frm ther tuttors

Hi Mohit,

I am not sure what you are referring to. You are directly contacting me now :).

I am really having a tough time with GRE.I have given Gre twice.First time i did not get the right direction to work and my maths is horrible.I had even joined classes for GRE but that did not help me out .I ended up with 291.And then again i started working with great hopes,i started with Nova maths bible.I did both the old and new edition .Then i did new barrons which has both english and maths.

Then i did manhattan word problems,algebra,geometry,and sentence equivalence and text completion and reading comprehension.I even did princeton verbal workout,i have done ETS official guide twice and i even did all the 3500 words of barrons six to seven times and i even did MAGOOSH videos twice for maths and once for english,i have seen and done lessons once. i have given all the kaplan MST tests and given all manhattan tests given the powerprep exam .

In all the exams i used to get between 300 and 310,in powerprep i got 315 in first exam then when i gave my real GRE exam i ended with 299.I am not understanding what should i do.I am having real tough time with maths.I am not able to get through it.Even after so much practice i ended up with 152 in maths not more.In english i got 147 even thats not a good score.I am not understanding wat to do.In maths i am not able to get through the problem though i try hard and my most weakest area is quant comparison,word problems .In tricky sums i always get entangled very badly.I have a huge problem of making silly mistakes now and then though i try hard to avoid them.In english i have problem in tackling text completions and reading comprehension .Please help me out i am in a big trouble.I need to get 310 and thats around 1200 according to new gre for my Phd course.1200 is the minimum requirement for Phd and i am not even getting around it.Please help me out i am determined to get it and i am ready to work for it.

I was really satisfied with Magoosh material its really very good the way each and every sum is being explained and all engish practice sums are explained.I need more help for maths and for english both. i hope u can give me a solution .I m planning to give my exam again in September first week so i have one month with me.Nothing is there which i have left to work out .I read all your blogs and i try to apply the tricks which you have mentioned ,but i m not getting where i am lagging behind.Please show me the direction .I will be waiting for your reply.

If possible give me a designed plan for my study as i already follow plans of ur blogs…

thanks…got to learn a new tip

You are welcome – glad it was helpful :).

the same problem could be attempted using weighted average also..right…

Yes, weighted average can be a very powerful, time-saving technique 🙂

Hi Chris:

Hope you are fine. I am planning to write GRE in October. I was going through the official guide and recognized that new GRE is placing more emphasis in statistics, data interpretation and probability questions. And many published prep materials in the market somewhat lack in that respect. Do the prep materials provided by magoosh adequately deals with these type of questions ? thanks

Yes Mezbah,

In fact we are just releasing a new batch of Data Interpretation questions. In general our products contain many questions on statistics, probability, and even permutations question. Unlike book-based test prep publishers, we are also at the advantage of being able to respond to students’ needs. If you (or anyone) think we are deficient in any question type, we are always able to answer that need by creating more questions.

Hope that helps 🙂