  ### Standard Deviation

Today on Maths is Fun, I’m going to talk about Standard Deviations.

That’s a really overly complicated term, that just earns the average distance away from the average that each data point is. And if you’ve ever heard the expression, I’m being graded on a curve, they’re using a Standard Deviation to do that.

So what they’re doing is they’re taking the average results of a test and they’re seeing how far away from everyone is from the average and then normalising the data so that people who might not have a passing mark might not have got over 50%. They normalise the data to say – well, how far away is everyone from the average, then they work it out that way. That’s what grading on a curve means.

And it’s telling you is that, that in a normal set of data like this here, how far away is all data from each other, and with a Standard Deviation, or you find that within minus one Standard Deviation either side of the average. You’ll find something like 68% of the data will fall into that bucket. And over here, if it’s within two Standard Deviations, either side is about 95%. And if you include the third Standard Deviation here is something like 99.7% of all data will sit somewhere inside of that. This huge amount of uses for Standard Deviations.

And let’s look at the formula because it’s kind of horrific. but it’s not too difficult if you can understand. So we’re going to go through with how this formula works. But the symbol for Standard Deviation is weird, a white circle with a little lip on it equals the square root of Sum Xi minus Xbar squared over the number of values. So it’s a pretty horrible looking formula. But we’re going to work it out right here on this whiteboard, because it’s actually really, really easy.

So over here, I’ve got some data. You can see here, I’ve got a range of values. And you can see the average of that is eight. So what I’m going to do is, I’m going to look at what the distance is, of each of these values is from that average.

So you can see here, this is a nine. So the distance here is one, this is an 11. So the distance here is three, this here is 13. So the distance here is a five, this here is 10. So the distance from the average is two, this one here is five. So the average this isn’t here is minus three, this is a two, so it’s minus six. This is a six, so it’s minus two, and then this is zero because it’s on the average.

So we’re going to do is we’re going to plug all of these values into this formula here because that’s what this bit of the formula means like for every X value, which is every point I’ve got. That’s what the X pi means. I’m going to minus it from the average. So this is X bar means average. There’s a couple of different symbols for average, which is the one I’m using.

So I’m going to take away each value from the average. I’ve just done that. Now, why am I squaring it? Well, that’s just a really simple trick, actually, in Maths, if we want to get rid of a negative, the easiest way to do that is to square it. And then square root, which is exactly what this formula is doing. Here we have a square just to get rid of the negatives. And then we’re going to square root it to bring that data back to the right number. And the reason we have to do this is I’m going to add all these together. And if I add positives and negatives that kind of cancel each other out. I don’t want them to cancel each other out. because I’m not really interested that this is negative three. I’m more interested that it is three away from the average. This is three away from the average.

See these two would cancel each other out, but they shouldn’t because they’re both three away from the average. So I’m going to do is I’m going to plug all these values in. So we’re going to do this bit of the formula here. We’re going to already minus that, we’re going to square each of them. So over here I’ve got one squared is one plus three squared, which is nine plus five squared, which is 25 plus two squared, which is four plus-minus three squared, which is 9 plus-minus 6 squared, which is 36, plus-minus 2 squared, which is 4, plus 0 squared, which is 0. And I’m adding them all together.

That’s what this formula here means. This just means sum. So we’ve already done this bit of the formula here.

Now we’re going to divide it by the number of data points we have, which would be 1, 2, 3, 4, 5, 6, 7, 8. So we’re going to divide that by 8. And then we need to square root the answer because we’ve squared all these values in here, so we need to bring it back to what it should look like. So that then equals 88. So that’s 88 over 8, square root, that equals 11. Square root mat equals 3.3.

So we’ve just done this really complicated formula and we’ve just had a look at this data here and it’s turning.

It’s telling us that all of these data points, the average distance, each data point is away from the average itself is 3.3. So one Standard Deviation is going to be 3.3 above 8, and 3.3 below 8, which we actually put on here, so we can go its 8, 9, 10, 11, 3. So that’s one Standard Deviation there. And we do the same here at about there. So that’s one Standard Deviation there. and you can see that probably about 68% of those data points are falling within there.

The second Standard Deviation would be somewhere up here, and the third one would be down here and up here.

So that Standard Deviations is really not hard, but you just need to understand the language of Mathematics. It looks like a really complicated formula, but it really isn’t.

I hope you’ve enjoyed the video.