How To Calculate Variance

Hi, my name is Charles and I'm one of the maths teachers from the Maxim Workshop. I'm just going to now teach you how to do some math. Hi.

In this video, I'm going to show you how to calculate variance which is a quantity that is measured for our statistics. Whenever you have a probability distribution, variance is something that shows you a measure of how far your variables are associated to that distribution and away from your expected value, and your expected value is mainly your mean value, your average. Okay, so variance is denoted by Var(x), so variance and the distribution involves quantities relating to x, so that's your variable.

Now, the basic equation that you might use to calculate variance is this one here, you've got your mean value subtracting your x values, so that would be your measurement of height, weight, mainly any variable that you're studying, and then you square this, and then the last term in this equation is your n and that's how many quantities you have relating to x. So just a brief example of what you might find is say, if you wanted to calculate the variance in various heights of human beings, let's say you have a measurement. So x, your distribution, and you have quantity relating to height as 7, 8, 8, 9, 7, and 6, now just imagine if these are your measurements for, say heights of children in a classroom.

Now, the next thing you would want to do is sum these so you can obtain the expected value okay, which is your mean. Now this is 15, this 8 plus 15 is 23, now 9 plus 23 is going to be 32, so we've got that and just shift it over here. And 32 plus 7 now is going to be 39, and 39 plus 6 is going to be 45.

So if you look at this, in our distribution, we have 1 2 3 4 5 6 values, so we're trying to divide 45 by 6. Now in order to do that, now we've got 45 divided 6. 6 doesn't enter 4, so we put 0 there, cross that off.

In our 6 times table, we got to 7 times 6, 42, and we've got there, we've got a remainder of 3, so we put the 3 there, now we want to see how many times 6 enters 30. So 6 x 6 is 36, but obviously we drop down to 6 x 5, so now we put a decimal point up here, and that is sufficient enough to take as our value for mean. So 7.

5, now what we want to do is stick this value inside here and actually calculate this difference here that we've got with squared then. 7, 8, 8, 9, 7, 6 and our mean value which is our expected value is basically 7.5.

Now what we're going to have to do now is stick these distribution values for x which is the height of kids and our expected value which is 7.5 into the equation and to see what we get. Okay, so what we're going to do now is stick these values into our equation with our expected value.

Now, what we should have is a summation here of all our expected value subtracting the height values and square it. We've got 7.5, take away 7, that's just 0.

5 squared, and we've got 7.5, take away an 8 which is negative 0.5 squared which would be just the same thing as this, and again we've got the same thing here, so plus (-0.

5) squared. Now, we'll come down here. Now, we've got 7.

5, take away 9 which would just be 1.5 squared, a (-1.5) squared, and we've got 7.

5 take away 7, which would again just be 0.5 squared. We also have 7.

5, take away 6 which would be 1.5 squared. Okay, so all of these added up will give us a variance and obviously divide it by n.

We've got a half squared, so that's just a half and a half which is a quarter, so that is 0.25 plus again 0.25, we've got again 0.

25. Now if you think again about 1.5, you can say it to yourself 15 squared would be 2.

5, so that's just pretty much going to be 2.25. And again, you've got 0.

25 again you've got 2.25. Now I can sum quickly this with this to make 4.

5, and I can sum these 4, obviously this together will make 4.5, this together will make 0.5, so that is just going to be 1.

So we just basically got 1.5 which is 5.5 and if you look at how