In this video, we're going to talk about weighted averages.
We'll talk about what a weighted average is, what a weighted average means and how to calculate
a weighted average.
Here's an example where we calculate a weighted average.
Let's say that we have a group of kids here, there are five kids, some boys and some girls
and for simplicity sake let's just say that each one weighs a hundred pounds.
Now one of their father shows up and the father is a bodybuilder.
Check out these muscles, check out these eight pack abs, I guess this means the guy has his
shirt off which is kind of creepy but anyway he's a big body builder and he weighs 300
pounds.
So now we have a group of six people here.
The kids 100 pounds each and the father 300 pounds.
So now you can ask a question about this group of people, you could say what is that average
weight?
We got these two different weights here, 100 pounds and 300 pounds, so you could calculate
what I call a regular average.
You take 100 pounds and 300 hundred pounds because those are the two choices and you
add them together and divide by two because they're two different weights.
You get 200 pounds and 200 pounds is right smack dab in between 100 and 300.
So that's one way that you could get the average weight, 200 pounds.
But here's the thing, is it really fair to say that the average weight of this group
is 200 pounds because there are five kids that each weigh 100 pounds and there's only
one father that weighs 300 pounds.
So to say that the average weight is right in the middle of those two weights kind of
doesn't make a lot of sense.
It feels like we should be able to take into account the fact that there are many more
kids who weigh a lot less than the one bodybuilder father.
This is where the idea of a weighted average comes into play.
A weighted average takes into account how many things in each group you have.
Here's how I would calculate the weighted average in this group of people, okay?
I would take the fact that there are five kids here that weigh 100 pounds each so I
do 100 + 100 + 100 + 100 + 100, that's five hundreds for each of the five kids and then
I add 300 for the weight of the bodybuilder and I divide by six because there are six
things total.
Sometimes it's easier to express this with multiplication, I've got five times a hundred
plus one times three hundred divided by six because there are six things and when we crank
through these equations we end up with a different answer than this, we end up with the 133 pounds.
This is the answer to the weighted average.
Now you'll notice this number 133 is much less than 200 and 133 is a lot closer to the
weight of the kids.
That's because of the weighted average and how it works.
The weighted average will pull down the average closest to whatever we have the most of, okay?
So if we're just half and half it would be 200 pounds, we would have the same number
of each, but since we have more of the kids we take that 200 hundred and we pull the number
down closer to the weight of what we have more of.
So that's why the weighted average for here is a 133, very close to the weight of the
kids.
Now the weighted average could work the other way as well.
Let's say that we had a group where four people were 300 pound bodybuilders and there's only
one kid that weighed 100 pounds.
In that case, we calculate the weighted average.
Four 300 pound bodybuilders plus one 100 pound kid divided by five total or we could do this
with multiplication (4x300) plus (1x100) divided by five and in that case we get 260 pounds,
alright?
This number is a lot higher than the number 200 that's right in between and it's much
closer to the weight of the bodybuilders.
It's much closer to 300 because they're more of the body builders and there's only this
one kid.
So the weighted average should be closer to the weight of what we have more of.
Now for weighted average, we don't just have to have two things like the kids and the bodybuilders,
we can have a number of different things more than two and we can have a whole bunch of
them.
In this case I'm talking about a parking lot that has three different kind of cars.
They're all called Lemonas because they look like lemons.
We got the Lemona G that weighs 3,000 pounds, the Lemona GX that weighs 4,000 pounds and
the Lemona GXL which weighs 5,000 pounds.
I have a different amounts of each of them.
So how would I calculate the weighted average here?
What I do is I take the weight of Lemona G, that's 3000 pounds, and I multiply it by how
many Lemona G's there are.
They're 32 times 3000 pounds each, I take that and then I add the number of GX's.
So I've got 5 of them, 5 times 4,000.
And then the GXL's, I've got 7 of them, 7 times 5,000 pounds.
Now I do this, I've accounted for the weights of each one of these cars and now I've got
to divide by the total number of cars I have in this group and that's 44.
So these are the G's, these are the GX's, and these are the GXL's.
Do the multiplication of each, add it up, and I'll end up with 3,432 pounds.
Now look at how this number compares to weights of the individual cars.
This number is closest to the weight of the G and that makes sense because I have a lot
more of the Lemona G's than I have of the heavier GX's and GXL's.
So this is the weighted average where as the regular average for these cars would have
been 3,000 plus 4,000 plus 5,000 divided by 3 which would have given us a number that
was smack dab in the middle.
But once again the weighted average here gives us a number that takes into account how many
of each type of these cars I have and gives me a number that is much closer to the type
of car that I have the most of.
Now a lot of times when you're talking weighted average, you'll have to work with percent
so instead of getting the number of things you'll get the percentages of them like right
here.
Here's how you calculate a weighted average using percent.
Okay, what I'm going to do is I'm going to take the weight of the Lemona G here and I'm
going to multiply it by 73 percent expressed as a decimal.
So the decimal place in 73 percent would be here.
To turn this percentage into a decimal I'll have to move the decimal place two spots to
the left so I'm going to move it to here to here, okay?
The decimal place is going to end up right here.
So I'm going to do 0.73 and then multiply that times the weight, 3,000 pounds, and that's
the Lemona G.
Now for the GX, I'll take 11 percent and express that as a decimal.
Move the decimal place two spots the left so it's going to end up here, 0.11 and multiply
that times the weight of the GX, 4,000 pounds for that model the car.
And finally I'll take 16 percent and express it as a decimal, 0.16 and multiply it by the
weight of the GXL model.
Now when you're working with percentages, you don't divide by anything.
All you do is multiply the weights by the percentages expressed in decimals and that's
all you have to do.
This is the same problem as the one that I did before, we're just expressing these abundances
or the amount that we have in percent so the answer 3,432 pounds is going to be the same
as what we got earlier.
Anyway, that's how you do weighted average using percent.
Now I have to say this, weighted average doesn't mean that you have to use weights of things,
alright?
You can calculate a weighted average with the amount of money people make or like how
much volume you have and different types of containers, anything you can think of, any
measurement you can think of you can do a weighted average of.
So I've been using examples of weights here, weights of people, weights of cars but you
don't just have to do weights for a weighted average.
The only reason you call it a weighted average is because it takes into account the number
of each thing you have and they kind of pull the average closest to their value so that's
why you call it a weighted average because you give different weight to each type of
thing that you have.
Finally, if you're interested to see how this equation here is the same as the one that
I did here, you can finish this video up and I'll show you at the very end how these are
equivalent expressions.
So these two expressions are equivalent ways of writing the calculations for the weighted
average for this information here.
Let me show you how they're equivalent.
We're going to take this and we want to end up with something that looks like this.
The first thing we can do is we can see these three different things that are added together
on the top of a fraction.
We can separate them out, okay?
So, I'm going to do 32 times 3,000 divided by 44 plus 5 times 4,000 divided by 44 plus
7 times 5,000 divided by 44, okay?
So I can split this into three pieces, each of them have the same denominator.
Now take a look at this.
I've got a number over 44 in each one of these cases and this is kind of a percentage because
44 is my total, and each one of these numbers in the numerator are fractions of that total.
So what I can do since these are multiplied together is I can divide this and then just
keep it multiplied by that.
So 32 divided by 44 gives me 0.73, that's this part, times 3,000 (you can put that in
parentheses).
Now I can do 5 divided by 44 and that gives me 0.11 and I'm going to multiply this now
times 4,000, put that in parentheses, and finally 7 divided by 44 gives me 0.16 times
5,000 and check it out!
What I end up with when I split these up and I do the division is exactly the same as the
number that I get when I take these values and use the straight percentages here.
So the point is whether you choose to use the individual numbers of each thing you have
or whether you choose to use the percentages, the way you go about calculating a weighted
average has the same math in it.
Anyway, now that you know how to calculate a weighted average using the numbers of the
things or the percentages of the things, you can now go on and learn how to calculate atomic
mass.