What is a Weighted Average? - YouTube

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.