What is a Weighted Average?

In this video, we're going to talk about

weighted

average

s. We'll talk about

what

a

weighted

average

is,

what

a weighted average means, and how to calculate a weighted average. Below is an example where we calculate a weighted average. Let's say we have a group of kids here, there are five kids, some boys and some girls, and for simplicity's sake, let's say they each weigh a hundred pounds. Now one of his parents appears and the father is a bodybuilder. Look at these muscles, look at these eight abs, I guess this means the guy is shirtless, which is a little creepy, but anyway he's a huge bodybuilder and weighs 300 pounds.

Now we have a group of six people here. The children 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 have these two different weights here, 100 pounds and 300 pounds, so you can calculate what I call a regular average. You take 100 pounds and 300 hundred pounds because those are the two options and you add them and divide them by two because they are two different weights. You get 200 pounds and 200 pounds is right between 100 and 300. So that's one way to 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 children who each weigh 100 pounds and there is only one parent who weighs 300 pounds? So saying that the average weight is right in the middle of those two weights doesn't make much sense. It seems like we should be able to take into account the fact that there are many more children who weigh much less than the bodybuilder parent. This is where the idea of ​​weighted average comes into play. A weighted average takes into account how many things you have in each group.

This is how you 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 bodybuilder's weight and I divide it by six because there are six things in total. Sometimes it's easier to express this with multiplication, I have five times one hundred plus one times three hundred divided by six because there are six things and when we look at 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 will notice that this number 133 is much smaller than 200 and 133 is much closer to the children's weight. This is because of the weighted average and how it works. The weighted average will reduce the average closest to whatever we have the most, okay? So if we're just half and half, it would be 200 pounds, we'd have the same number of each, but since we have more kids, we take those 200 pounds and put the number closer to the weight of what we have. have more than. That's why the weighted average here is 133, very close to the weight of children.

Now the weighted average could also work the other way around. Let's say we had a group where four people were 300-pound bodybuilders and there was only one kid who was 100 pounds. In that case, we calculate the weighted average. Four 300 pound bodybuilders plus a 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, okay? This number is much higher than the number 200 which is right in the middle and is much closer to the weight of bodybuilders. It's much closer to 300 because they're more bodybuilders and there's just this kid.

Therefore, the weighted average should be closer to the weight of what we have the most. Now for weighted average, we don't just have to have two things like kids and bodybuilders, we can have more than two different things and we can have a lot of them. In this case I am referring to a parking lot that has three different types of cars. They are all called Lemonas because they look like lemons. We have the Lemona G that weighs 3,000 pounds, the Lemona GX that weighs 4,000 pounds, and the Lemona GXL that weighs 5,000 pounds. I have different amounts of each of them.

So how would you calculate the weighted average here? What I do is take the weight of Lemona G, which is 3,000 pounds, and I multiply it by the amount of Lemona G there is. That's 32 times 3000 pounds each, I take that and then add the GX number. So I have 5 of them, 5 times 4000. And then the GXLs, I have 7 of them, 7 times 5000 pounds. Now I do this, I've tallied up the weights of each of these cars and now I have to divide them by the total number of cars I have in this group and that's 44. So these are the Gs, these are the GXs, and these are the GXL.

Do the multiplication of each one, add it up and I end up with 3,432 pounds. Now look at how this number compares to the 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 Lemona G than the heavier GX and GXL. So this is the weighted average, where the regular average for these cars would have been 3000 plus 4000 plus 5000 divided by 3, which would have given us a number that was right in the middle. But again, the weighted average here gives us a number that takes into account how many of each type of these cars I own and gives me a number that is much closer to the type of car I own the most.

Now, a lot of times when you're talking about weighted average, you're going to have to work with percentages, so instead of getting the number of things, you're going to get the percentages of them, like here. This is how you calculate a weighted average using a percentage. Well, what I'm going to do is take the weight of the Lemona G here and I'm going to multiply it by 73 percent expressed as a decimal. So the 73 percent decimal place would be here. To convert this percentage to a decimal I'm going to have to move the decimal place two points to the left, so I'm going to move it from here to here, okay?

The decimal place will end right here. So I'm going to do 0.73 and then multiply it by the weight, 3,000 pounds, and that's the Lemona G. Now, for the GX, I'm going to take 11 percent and express it as a decimal. Move the decimal place two points to the left so it ends here, 0.11, and multiply it by the weight of the GX, 4,000 pounds for that car model. And finally I'm going to take the 16 percent and express it as a decimal, 0.16, and multiply it by the weight of the GXL model. Now, when you work with percentages, you don't divide by anything.

All you have to do is multiply the weights by the percentages expressed in decimals and that's all you have to do. This is the same problem I did before, we are just expressing these abundances or the amount we have as a percentage, so the answer 3,432 pounds will be the same as we got before. Anyway, this is how you get the weighted average using percentage. Now I have to say this, weighted averaging doesn't mean you have to use weights of things, okay? You can do a weighted average with how much money people make or how much volume you have and the different types of containers, anything you can think of, any measurement you can think of you can do a weighted average.

So I've been using examples of weights here, weights of people, weights of cars, but you don't need to just do weights for a weighted average. The only reason you call it a weighted average is because it takes into account the amount of each thing you have and they somehow bring the average closer to its value, which is why you call it a weighted average because you give a different weight to each type. of thing you have. Finally, if you're interested in seeing how this equation here is the same as the one I made here, you can finish this video and I'll show you at the end how they are equivalent expressions.

So these two expressions are equivalent ways of writing the calculations here for the weighted average of this information. Let me show you how they are 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 look at these three different things that add up to the top of a fraction. We can separate them, okay? So, I'm going to do 32 times 3000 divided by 44 plus 5 times 4000 divided by 44 plus 7 times 5000 divided by 44, okay? So I can divide this into three parts, each of which has the same denominator.

Now look at this. I have a number greater than 44 in each of these cases and this is a type of percentage because 44 is my total, and each of these numbers in the numerator are fractions of that total. So what I can do, since these are multiplied, is divide this and then keep it multiplied by that. So 32 divided by 44 gives me 0.73, that's this part, multiplied by 3000 (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 by 4000, put it in parentheses, and finally 7 divided by 44 gives me 0.16 times 5000 and check it out!

What I get when I divide them and do the division is the exact same number that I get when I take these values ​​and use the direct percentages here. So the point is whether you choose to use the individual numbers for each thing you have or whether you choose to use the percentages, the way you calculate a weighted average has the same math. Anyway, now that you know how to calculate a weighted average using the numbers of things or the percentages of things, you can go ahead and learn how to calculate atomic mass.