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