Learn about Statistics: Variance of a Population

I realize I made a slight error in the last video when I talk about the population and the sample mean, so I will rewrite the equation, I realize I made a slight notational error and that might confuse you a little bit. So just a review little bit it’ll never hurts.

The mean of a population once again that’s mu, the mean of a population is equal to—you take the sum of each of the data points. So you take the sum, that’s that bug sigma’s four of each of data points, so X sub I, had I written X sub N before and if you review the last video you could see why I might be little confusing and you start with the first data point so I is equal to one, you start with the first data point and you take the sum all the way to the nth data point where we have a big capital N, where N is a total number of elements in the population and they divide that by N.

So, that’s another way of writing X sub one plus X of two plus and you just keep adding bam, bam, bam, however may they are X sub N and then you divide that by N and I think that’s what you’re with this just arithmetic mean or the average, you just add all of the elements and you divide by the total numbers of elements there are, that’s just the fancy way of writing that and then the sample mean is essentially the same thing although you slightly different notation, the sample mean is written as X with the line over it and that’s equal to, once again the sum of the elements in the sample and they have just slight notational difference.

You start at the first element in the sample and you go to the number of elements in the sample and that’s why they use that lower case N they are big N elements in the whole population and if you took some sub set of that, we assuming that N is less small N is less than our equal to big N.

Anyway, you divide that by the total number of elements in the population, oh in the sample, so once again this would be X1 + X2 + X lower case N divided by lower case N. These are essentially the same thing, if your sample was entire population hen these Ns would be equal to each other and this numbers would be equal to each other but just the notational deference, if you ever see this you know you are dealing with the sample here you know you’re dealing with the entire population and similar big N entire populations small N the sample.

I think we’re now ready to learn a little bit about measure of dispersion. So the mean and the mode and the median that we covered in the first video in this play list where all ways of measuring the central tendency of a data center, kind of picking a number that is most representative of all the numbers. But we loose a lot of information, we don’t know whether all the numbers in the data’s are as close to that number, close to the mean or maybe they are really far away from the mean and that’s way we want to come up with measures of dispersion, dispersion.

Let me tell you; let me show you what I mean. So let’s say I have one set and lets say, it a two, a two and a three and a three ad let’s say this is a population, let’s just deal with population, population means and population dispersion for now. So where’s the mean here, the mean here is is going to be 2 + 2 +3 + 3 all of that, all of that over four and what is that? That’s equal to 2 ½, 4 + 6 divided by four that’s equal to 2.5, fair enough.

Now, what if we had this, what is we had the numbers zero, zero and five and five, so these are that set up with commas, and just you know these are separate numbers. What’s the mean here? Well the mean here and let’s say this is the population that we’re sampling, well this is the samples, these entire population ad you’ll say why I’m making a distinction later. So it’ll be 0 + 0 + 5 + 5 well that’s ten divided by four is equal to 2.5.

So, the arithmetic means of both of these populations are the same number, they’re both 2.5. You see that this set such are kind of there different, here all of the numbers are pretty close to 2.5. While here, sure their mean, their arithmetic is 2.5 but they are further away from 2.5 or the distance of each of this numbers, each of the data, each of the numbers in the set, there distance from the mean is further so you can kind of view that there more disperse, they are further away from the mean or another way you could think of it is the mean although it is, it does measure its central tendency it doesn’t, it’s not quit as indict of all the numbers and numbers are much further away from the mean on average.

So, how do you measure that? Well, you measure that with a variants and this is something I found and even in my own, it seems complicated when you first look at it and, will statistic textbooks use fairly complex notation but the idea is almost just straight forward as the arithmetic mean.

So, what they’ll do is no right, the variants and the right is this letter sigma, this Greek letter—I put the top part too long let me actually undo that, I don’t want you to spend the rest of your life writing with the big top part—they write us sigma ^2 and we’ll talk that in a few seconds about Y is written as—why don’t they just write V for variants why do they write these weird letter square and we’ll talk that about in a second.

But the variants of a population is defined and once again these are just human derive constructs so of time to get our mind around data, be able to described instead of data without having to list all the numbers and being able to kind of understand what that data might represent or what kind a represent that data.

So, what you do is you take the sum, you take the sum and you start all of the points in the population but instead of taking the sum of the points you take each point, X sub I and you subtract from that and actually doesn’t matter I you subtract from that or subtract that from the mean, the population mean and then you square it.

So what is this, this is the distance between each number and the mean and then when you square it becomes a positive number so you can kind a view that just so the squared absolute distance between each number and the mean of that set and then you take the average of all of those. And you divide that by N. That might seem like a very complicated notion but let’s calculate it for these two data sets.
So, here—let me rewrite that first data set, it’s two, two, three and three. So what is—actually let me write it this way, I think this will help explain it to you for you a little bit better. So, if I wrote I, I1, I2, I3 ,I4, that’s its I then X sub I and you know it’s kind of arbitrary just you know this is saying the first term, the second, the third term, I could have this in any order it doesn’t matter. Maybe this was the first term and this is the second and this is the third, it doesn’t matter because we are just going to add them all up and then divide them so it doesn’t matter what order we do it.

But anyway, X sub one, X sub one is equal to two, X sub two is equal to two, X sub three is equal top three, I’ll stop writing this equal thing, X sub four is equal to four. What is the mean? Well, we figured out the mean up here, we just took these numbers and added them and divided by four, the mean is 2.5, the mean is 2.5.

So, what is X sub I minus the mean? We’re slowly building up to this equation. What is X sub I minus the mean? Well, 2 – 2.5, that’s minus 0.5, 2 – 2.5, that’s once again minus point five. 3 – 2.5 that’s .5, 3 – 2.5 that’s 0.5, fair enough.

Now, this equation they want us to square this so X sub I minus the mean squared and that has, of there several other properties we’ll talk about later but the most important thing to the squaring does and the absolute value could’ve done it as well but the squaring makes all of these positive so, minus 0.5 square this positive 0.25, this is positive 0.25 plus 0.5 will be also positive 0.25 and this is 0.25.

And so, what is—if we wanted to know the sum from I is equal to one to four of X sub I minus the mean which is 2.5^2 this is equal to what, the sum of all of these numbers, this is just saying sum all of these, so sum all o these 0.25, so that’s equal to one but this isn’t the variants yet, the variants is this thing—lets’ get the original formula. The variant is this thing divided by the total number of numbers you have.

So you take this and then you—so the variants is equal to this thing divided by the total number of numbers which is four which is equal to 0.25. And you see, I mean here are the distances from every number square, the distance from every number to mean squared was 0.25. So the average of all of this which is essentially what the variants is the average was also 0.25.

And then I’ll do another example where these are different, the other example of this video actually they’re not different. What you see here the average square distance from the mean and that first set is set as 0.25. And here what’s the average square distance from the mean? So, this is how far from the mean, it is—so let’s say X, let me write X sub I and then X sub I minus the mean for this population. So X sub I there is zero, there’s a zero there’s a five and a five, this a first term X sub one, X sub one, this is X sub two and so forth. And then each of these numbers minus zero, minus two plus 2.5, zero minus 2.5, this could be 2.5, that’s the mean. It’s -2.5, five minus 2.5 is 2.5.

Now, as you took X sub I minus the mean squared, 2.5 square is what, 6.25 becomes positive so 6.25 that’s the same thing 6.25, that’s already positive, so 6.25. And so the variants are a sum of all of these divided by the total numbers there are.

So we take the sum of all, so this is the average of this and that’s pretty easy to calculate, if you took, if you add all these up and divide by four, you’re just going to get 6.25. So, the variant of this population is 6.25. So there you have it, you have two data sets where there means are the same but the variants of this data set is equal to, we figure out it was 0.25 while the variants of this data set is equal to 6.25.

And it’s hard right to have intuition of what this, what the six related to 0.25 but you know that this is a larger number; this is a much larger number. Then this is, which tells you just kind of an intuitive feel that the numbers in the set are on average much further away from the mean than the numbers in this data set.

Anyway, I’m out of time I’ll see you in the video and we’ll talk a little about this and I will talk and we’ll move in to the standard deviation then what happens if you take use of a sample instead of a population, everything we’re doing were taking the mean and the variants of every number in the data set, later will do it fro the sample. See you soon.