The phrase “weighted average” may be a little scary sounding, but it’s nothing to get freaked out over. Usually weighted averages on the SAT will use the basic formula for finding the mean (link to “SAT Math Types of Averages“). It’s pretty much the same skill.
What is a “weighted average”?
Basically, weighted means uneven, here; the numbers that you’re looking at don’t carry the same importance. For example, if I’m trying to find the average number of fleas that my pets have, and each cat has 150 while each dog has 200, then those two numbers have equal “weight” only if I have the same number of cats as dogs. Let’s say I have 1 of each.
That’s just a normal mean, so that’s no problem. Well, the fleas are a problem, I guess. And the fact that I’m counting fleas might have my family a little worried…anyway, the math is easy. But that’s a non-weighted average.
For a weighted average, I would have a different number of cats than dogs. Let’s say I had 3 cats and 2 dogs. (And they all have fleas…things are starting to get kinda gross. Sorry.)
In order to give them the appropriate weight, we’d have to multiply each piece appropriately and change the total (denominator) to reflect it.
But if you expand that, you’ll see that it’s the same as the standard mean formula.
Just make sure you divide by five (because I have seven pets) not two (for two types of pets).
Finding average rates
Average rates are a type of weighted average. Your SAT will include a problem or two about these, and you need to be sure not to fall for the common trap.
Maria’s drive to the supermarket takes her 20 minutes, during which she averages a speed of 21 miles per hour. She takes the same route home, but it only takes 15 minutes to cover the equal distance. What was Maria’s average speed while driving?
- 15.5 mph
- 21 mph
- 24 mph
- 24.5 mph
- 28 mph
This is a tricky, multi-step problem, and you can’t plug in the answer choices to solve it, sadly.
Let’s first find all of our information, because the question has only given you part of it. You need to know the formula r=d/t (rate = distance/time), also expressed as d=rt (easily remembered as the “dirt” formula). We’re going to use it both ways.
Using that formula, let’s look at the first leg of her trip. She travelled for of an hour at 21 mph, so she must have travelled 7 miles.
Using that info, we can figure out the rate of her trip back home. Going 7 miles in of an hour on the way home, she went an average of 28 mph.
So now we need to find the total average. That’s not the average of the two numbers we have! Because each mile she travelled on the way there took more time than each mile on the way home, they have different weights!
Instead, you need to take the total of each piece—total time and total distance—to find the total, average rate.
Weighted averages that you won’t see on your SAT
I’ve never seen an SAT question that asks you to find an average based on percent weights (e.g. finding a final grade in a class where quizzes count for 70%, attendance for 20%, and participation for 10%). Finding that average is a little more complicated, so it’s nice that we don’t have to worry about it.
If you’re finding the average of two sets of information that already are averages in their own right, like the number of fleas per cat and the number fleas per dog, you can’t just take the mean of those averages. You have to find the totals and then plug them into the formula. You should be excited for these kinds of problems, if for nothing more than having the opportunity to bust out your handy-dandy, brand-spankin’ new SAT calculator. 😛