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• Let's review a little bit of everything we learned so far

• and hopefully it'll make everything fit together

• a little bit better.

• Then we'll do a bunch of calculations with real numbers

• and I think it'll really hit the point home.

• So, first of all if we're dealing with a-- let me

• actually write down, let me make some columns.

• So if we're dealing with-- let's see, we could call it the

• concept and then we'll call it whether we're dealing with

• a population or a sample.

• So the first statistical concept we came up with was the

• notion of the mean or the central tendency and we learned

• of that was one way to measure the average or central

• tendency of a data set.

• The other ways were the median and the mode.

• But the mean tends to show up a lot more, especially when we

• start talking about variances and, as we'll do in this video,

• the standard deviation.

• But the mean of a population we learned-- we use the greek

• letter Mu-- is equal to the sum of each of the data points

• in the population.

• That's an i.

• Let me make sure it looks like an I.

• So you're going to sum up each of those data points.

• You're going to start with the first one and you're going

• to go to the nth one.

• We're assuming that there are n data points in the population.

• And then you divide by the total number that you have.

• And this is like the average that you're used to taking

• before you learned any of the statistics stuff.

• You add up all the data points and you divide by

• the number there are.

• The sample is the same thing.

• We just use a slightly different terminology.

• The mean of a sample-- and I'll do it in a different

• color-- just write it as x with a line on top.

• And that's equal to the sum of all the data

• points in the sample.

• So each of the xi in the sample.

• But we're serving the sample is something

• less than a population.

• And then you go to the lower case n where we assume that

• lowercase n is less than the big N.

• If this was the same thing then we're actually taking the

• average or we're taking the mean of the entire population.

• And then you divide by the number of data

• You get to n.

• Then we said OK, how far-- this give us the central tendency.

• It's one measure of the central tendency.

• But what if we wanted to know how good of an indicator this

• is for the population or for the sample?

• Or, on average, how far are the data points from this mean?

• And that's where we came up with the concept of variance.

• And I'll arbitrarily switch colors again.

• Variance.

• And in a population the variable or the notation for

• variance is the sigma squared.

• This means variance.

• And that is equal to-- you take each of the data points.

• You find the difference between that and the mean that

• you calculate up there.

• You square it so you get the squared difference.

• And then you essentially take the average of all of these.

• You take the average of all of these squared distances.

• So that's-- so you take the sum from i is equal to 1 to

• n and you divide it by n.

• That's the variance.

• And then the variance of a sample mean-- and this was a

• little bit more interesting and we talked a little bit

• about it in the last video.

• You actually want to provide a-- you want to estimate the

• variance of the population when you're taking the

• variance of a sample.

• And in order to provide an unbiased estimate you do

• something very similar to here but you end up

• dividing by n minus 1.

• So let me write that down.

• So the variance of a population-- I'm sorry, the

• variance of a sample or samples variance or unbiased sample

• variance if that's why we're going to divide by n minus 1.

• That's denoted by s squared.

• What you do is you take the difference between each of the

• data points in the sample minus the sample mean.

• We assume that we don't know the population mean.

• Maybe we did.

• If we knew the population mean we actually wouldn't have to do

• the unbiased thing they were going to do here in

• the denominator.

• But when you have a sample the only way to kind of figure out

• the population mean is to estimate it with sample mean.

• So we assume that we only have the sample mean.

• And you're going to square those and then you're going to

• sum them up from i is equal to 1 to i is equal to n because

• you have n data points.

• And if you want an unbiased estimator you divide

• by n minus 1.

• And we talked a little bit before why you want this to be

• a n minus 1 instead of a n.

• And actually in a couple of videos I'll actually

• prove this to you.

• One, I'll prove it maybe experimentally using Excel and

• then I'll-- which wouldn't be a proof, it'll just give you a

• little bit of intuition-- and then I'll actually prove

• it a little bit more formally later on.

• But you don't have to worry about it right now.

• The next thing we'll learn is something that you've probably

• heard a lot of, especially sometimes in class, teachers

• talk about the standard deviation of a test or-- it's

• actually probably one of the most use words in statistics.

• I think a lot of people unfortunately maybe use it or

• maybe use it without fully appreciating everything

• that it involves.

• But the goal we'll eventually hopefully appreciate

• all that involves soon.

• But the standard deviation-- and once you know variance it's

• actually quite straightforward.

• It's the square root of the variance.

• So the standard deviation of a population is written as sigma

• which is equal to the square root of the variance.

• And now I think you understand why a variance is written

• as sigma squared.

• And that is equal to just the square root of all that.

• It's equal to the square root-- I'll probably run out of

• space-- of all of that.

• So the sum-- I won't write at the top or the bottom, that

• makes it messy-- if xi minus Mu squared, everything over n.

• And then if you wanted the standard deviation of a

• sample-- and it actually gets a little bit interesting because

• the standard deviation of a sample, which is equal to the

• square root of the variance of a sample-- it actually turned

• out that this is not an unbiased estimator for this--

• and I don't want to get to technical for it right now--

• that this is actually a very good estimate of this.

• The expected value of this is going to be this.

• And I'll go into more depth on expected values in the future.

• But it turns out that this is not quite the same

• expected value as this.

• But you don't have to worry about it for now.

• So why even talk about the standard deviation?

• Well, one, the units work out a little better.

• If let's say all of our data points were measured

• in meters, right?

• If we were taking a bunch of measurements of length then

• the units of the variance would be meter squared.

• right?

• Because we're taking meters minus meters.

• This would be a meter.

• Then you're squaring.

• You're getting meters squared.

• And that's kind of a strange concept if you say you know the

• average dispersion from the center is in meter squares.

• Well first, when you take the square root of it you get

• this-- you get something that's again in meters.

• So you're kind of saying, oh well the standard deviation

• is x or y meters.

• And then we'll learn a little bit it if you can actually

• model your data as a bell curve or if you assume that your data

• has a distribution of a bell curve then this tells you some

• interesting things about where all of the probability of

• finding someone within one or two standard deviations

• of the of the mean.

• But anyway, I don't want to go to technical right now.

• Let's just calculate a bunch.

• Let's calculate.

• Let's see, if I had numbers 1, 2, 3, 8, and 7.

• And let's say that this is a population.

• So what would its mean be?

• So I have 1 plus 2 plus 3.

• So it's 3 plus 3 is 6.

• 6 plus 8 is 14.

• 14 plus 7 is 21.

• So the mean of this population-- you sum up

• all the data points.

• You get 21 divided by the total number of data

• points, 1, 2, 3, 4, 5.

• 21 divided by 5 which is equal to what?

• 4.2.

• Fair enough.

• Now we want to figure out the variance.

• And we're assuming that this is the entire population.

• So the variance of this population is going to be equal

• to the sum of the squared differences of each of

• these numbers from 4.2.

• I'm going to have to get my calculator out.

• So it's going to be 1 minus 4.2 squared plus 2 minus 4.2

• squared plus 3 minus 4.2 squared plus 8 minus

• 4.2 squared plus 7 minus 4.2 squared.

• And it's going to be all of that-- I know it looks a little

• bit funny-- divided by the number of data points we

• have-- divided by 5.

• So let me take the calculator out.

• All right.

• Here we go.

• Actually maybe I should have used the graphing

• calculator that I have.

• Let me see if I can get this thing-- if I could get this.

• There you go.

• Yeah, I think the graphing one will be better because

• I can see everything that I'm writing.

• OK, so let me clear this.

• So I want to take 1 minus 4.2 squared plus 2 minus 4.2

• squared plus 3 minus 4.2 squared plus 8 minus 4.2

• squared, where I'm just taking the sum of the squared

• distances from the mean squared, one more, plus

• 7 minus 4.2 squared.

• So that's the sum.

• The sum is 38.8.

• So the numerator is going to be equal to 38.8 divided by 5.

• So this is the sum of the squared distances, right?

• Each of these-- just so you can relate to the formula-- each

• of that is xi minus the mean squared.

• And so if we take the sum of all of them-- this numerator is

• the sum of each of the xi minus the mean squared from

• i equals 1 to n.

• And that ended up to be 38.8.

• And I just calculated like that.

• I just took each to the data points minus the mean

• squared, add them all up, and I got 38.8.

• And I went and divided by n which is 5.

• So this n up here is actually also 5.

• Right?

• And so 38.8 divided by 5 is 7.76.

• So the variance-- let me scroll down a little bit-- the

• variance is equal to 7.76.

• Now if this was a sample of a larger distribution, if this

• was a sample-- if the 1, 2, 3, 8, and 7, weren't the

• population-- if it was a sample from a larger population,

• instead of dividing by 5 we would have divided by 4.

• And we would have gotten the variance as 38.8 divided by n

• minus 1, which is divided by 4.

• So then we would have gotten the variance-- we would have

• gotten the sample variance 9.7 if you divided by n

• minus 1 instead of n.

• But anyway, don't worry about that right now.

• That's just a change of n.

• But once you have the variance, it's very easy to figure

• out the standard deviation.

• You just take the square root of it.

• The square root of 7.76-- 2.78.

• Let's say 2.79 is the standard deviation.

• So this gives us some measure of, on average, how far

• the numbers are away from the mean which was 4.2.

• And it gives it in kind of the units of the

• original measurement.

• Anyway, I'm all out of time.

• I'll see you in the next video.

• Or actually, let's figure out-- we said if this was a sample,

• if those numbers were sample and not the population, that

• we figured out that the sample variance was 9.7.

• And so then the sample standard deviation is just going to