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We're devoting this entire section to talking about the normal distribution because it's such an important

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distribution for modeling real world phenomena and because it's so important because it comes up so

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often, we're going to break it down into several pieces here.

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We just want to talk about the mean variance and standard deviation that are associated with a normal

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distribution.

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So as a reminder, a normal distribution is a distribution whose probability density function looks

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like this.

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So if the probability density function is F of x, then this curve here is modeled by this function

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F of x because of the shape of the normal distribution.

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We often call it a bell curve as well, although there are other distributions that have bell shapes.

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So this particular kind of bell shape is not exclusive to a normal distribution, but all normal distributions

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do have this kind of symmetrical bell shaped curve or bell shaped probability density function, which

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is centered at the mean of the data, which if we're talking about a population, we indicate with MU,

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if we're talking about a sample, we indicate with x bar.

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And that's right at this point right here, the center of the normal distribution.

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Now when we're talking about a normal distribution, we have to keep in mind that we could still be

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referring to either a population or a sample.

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Remember that a population is the entire group or set that we're studying, whereas a sample is a small

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subsection of that population.

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And we distinguish between these two things because let's say, for instance, we're interested in surveying

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the entire population of the state of California.

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We know right away it's going to be very difficult, if not impossible, to collect survey responses

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from every single person, every single citizen of the state.

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And so instead of attempting to ask every person in the state the same survey question or to gather

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data from every person in the state, the entire population, instead we'll take a small sample.

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The goal, of course, is to make sure that our sample is representative of the population, but that's

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an entirely different topic.

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Figuring out how to take a sample that isn't biased but instead is representative of the population.

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Now, for a normal distribution, our formulas for mean variance and standard deviation will be different

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depending on whether we have data from the entire population or just from the sample.

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Of course, keep in mind that normally we'll have sample data because practically speaking for real

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world problems, it is usually difficult or impossible to collect data for the entire population.

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But to distinguish between these two groups, we want to indicate the size of the population with capital

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N and the size of the sample with lowercase n.

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So if you see capital N, that means we're talking about the number of subjects in the entire population.

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So if we're talking about the state of California, this capital N would represent the number of people

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in the entire state.

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If we see lowercase n, it means that we've taken a smaller sample of the population and the number

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of people in our sample is given by lowercase n.

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So let's say based on what we're studying, not related to the California example, our population is

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10,000.

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Well, capital N would equal 10,000.

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We might take a sample of, let's say 1200 people and then lowercase n would be equal to 1200.

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So these are the sizes of the population in the sample capital n and lowercase n, And then the mean

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of the population is given by this formula here, whereas the sample is given by this formula.

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Notice that they are identical, except for two things.

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First, in the population mean formula, we're dividing by capital N, whereas in the sample mean formula,

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we're dividing by lowercase n, And of course that makes sense based on the fact that we're using capital

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n and lowercase n to represent the size of our population and sample respectively.

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And then the only other difference is the notation that we use to indicate the mean, which we talked

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about earlier up here as the center of the normal distribution.

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The population mean, we indicate with this Greek letter mu and the sample mean we indicate with x bar

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this x with a bar over it.

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But other than that, the formulas are identical.

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So all these formulas are telling us is that to define the mean, we are summing up all of the possible

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x sub values, all of the values that are variable takes on and after we sum them all up, then we divide

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by either the number in our population.

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If we're finding population mean or the number in our sample, if we're finding sample mean.

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So add up all the data points, divide by the number of data points we have.

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That's all we're doing to calculate the mean of the normal distribution.

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Now for the variance, this should look familiar because we've already looked at mean variance and standard

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deviation formulas.

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But for the normal distribution, just to confirm here, all we're doing with the variance is.

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Taking all of the data points that we have, all of the sub values that we have in our data set.

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And for each one we subtract the mean here.

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So notice that the formulas for population in sample are again almost identical.

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All we're doing is taking the individual x abi values in our data set and that goes for population and

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sample and then we subtract the mean.

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And of course if that's the population we're subtracting MU if it's the sample, we're subtracting X

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bar.

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But in either case we're just subtracting the mean and that gives us the deviation for each data point.

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So this is the deviation of the individual data point x by when we square that value, we now have a

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squared deviation or a squared difference.

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And then this summation notation just tells us to sum up all the squared deviations.

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So once we expand out to this part of the formula here, this is just the sum of all the squared deviations.

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And again, notice that up to that point our formulas are identical, except for the mean being represented

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by MI or X bar.

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And then in the summation notation for population, we're summing up to capital N for the sample, we're

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summing up to lowercase N, So really no difference at all, practically speaking, between the two

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formulas.

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Up to this point, we're just taking the sum of all of the squared deviations.

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But then for population, we divide by the number in our population capital.

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N, But for the sample, you might think that we would divide by lowercase n the number of subjects

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in the sample.

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But in fact, and this is the most interesting part of all of our formulas for mean variance in standard

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deviation for a population in a sample.

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This particular piece right here is the most interesting part of our formulas, and the reason is because

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we would think it's n, but it's actually n minus one.

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And that's because when we take a sample from a population, we inherently introduce some bias.

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It's virtually impossible for our sample to perfectly represent our population for us to be able to

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scale up our sample data to perfectly match our population.

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For example, maybe we're working with a population of people and let's say the population is 1000 people,

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and we'll even say that exactly 500 of them are women and 500 of them are men.

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We might take a sample of 100 people, and even though we randomly sample from the population, meaning

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that we are randomly, blindly choosing one person at a time from the population, putting them back

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in the population, and then pulling out another person.

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And we're collecting a sample of 100 people from that population of 500 women and 500 men.

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When we sample that way and we collect 100 people, we might get a sample that's, let's say 52 women

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and 48 men.

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That's not going to scale up perfectly to 500 women and 500 men.

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Our sample is going to be a little bit different in terms of its representation of the population.

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We are just hoping and implementing some good sampling techniques to try to make sure that the sample

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is as representative of the population as we can possibly make it.

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But no matter what we do, we're likely going to introduce bias.

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And so because we introduce some bias when we sample from the population, it turns out that because

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of that bias, using this end minus one figure to calculate variance actually helps us undo a little

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bit of that bias as opposed to just using lowercase n the number of subjects in our sample.

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So we use this n minus one value to get a better approximation to get rid of some of the bias we introduced

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by sampling instead of using the full population.

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So when we calculate variance for a sample, we have to remember to divide by n minus one instead of

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just lowercase n the number of subjects in our sample.

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So if we take a sample of 100 people, we would be dividing here by 99 instead of dividing by 100.

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So those are our population and sample variance formulas.

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And then, as you already know, standard deviation will always just be the square root of variance.

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Now, if we want to see these formulas in action, of course the easiest way to do this is going to

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be to use a calculator or a computer to help us make these calculations, especially as our dataset

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gets larger.

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Because you can see here, let's work with a data set that I've already created.

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This data set only has ten data points in it and already look at all of the calculations we're doing,

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much less if we have a data set of 1000 subjects or even more, these calculations quickly become impossible

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to do by hand.

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But here's what we've done here to calculate mean variance and standard deviation for this particular

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data set.

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Let's walk through this step by step.

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So what we're going to say is that maybe we're a distribution company.

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We're running a.

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Warehouse.

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And we want to get information about the age of the employees that work in the warehouse.

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So what we're going to do is we're going to take a sample of ten employees.

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So we're going to say lowercase n is equal to ten.

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We're going to be using a sample instead of surveying the entire population.

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Maybe we actually have 200,000 warehouse workers and we're not going to be able to ask every one of

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them for their age.

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So instead, we're going to take a sample of ten employees.

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And all we're doing here, we're not talking about at this point whether this is a good sample.

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We'll talk about sampling techniques and representative samples later.

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All we're talking about now is how to calculate mean variance and standard deviation for this sample.

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We're using age in this example only because age is as long as our population is large enough, is going

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to be normally distributed around some mean.

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So we're going to say here that our sample size is lowercase n equal to ten.

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We surveyed ten employees, and this second column here of our table are all of the responses we got.

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So we asked ten warehouse employees their age and we got all ten ages here.

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And then to calculate sample mean.

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So we're calculating here X bar we found that x bar the sample mean was equal to 43.2.

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And we find that by taking the sum of all of these ages and then dividing by N equals ten.

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So we would take here to get x bar the sample mean.

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This is how we calculated 43.2 we would take 43 plus 35 plus 38.

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So 43 plus 35 plus 38.

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And we would continue adding until we got to the very last age here 40.

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And then we divide that entire sum by the sample size and equals ten.

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And we see that here in our sample mean formula.

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We sum up all of our data points, all of our x sub data points, and then we divide by the sample size

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and equals ten.

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And the result there is a mean of 43.2 for the age of those ten employees.

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Now, to find variance for our sample, we take each one of our x sub values and we subtract the mean

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x bar.

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So that's what we're doing here in this column.

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We're finding all of the deviations.

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So we're taking 43 and subtracting 43.2 to get negative point two.

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We're taking 35 and subtracting 43.2 to get -8.2 all the way down until the last data point.

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And so this second column here is giving us this x sub minus x bar value for each data point.

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Then in this last column here, we're squaring each of those deviations.

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So -0.2 squared is a positive 0.0 for -8.2 when we square it is a positive 67.24.

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So we square all those values.

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Now we have all of our individual squared deviations.

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This part of the formula, then our summation notation here tells us add up all of those squared deviations.

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So we have all of our squared deviations and then we add them all together to find this sum.

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Now we have the sum of all of our squared deviations, and then all we have to do to find variance is

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divide this sum of all the squared deviations by and remember and minus one because we have a sample,

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not a population.

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And when we find sample variance, we have to divide by n minus one to undo some of that bias.

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So here this value for variance is not 1,007.6 divided by N equals ten, it's 1,007.6 divided by ten

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minus one or divide it by nine.

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So when we divide the sum of our squared deviations by n minus one equals nine instead of n equals ten,

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the result we get is 111.96.

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And so this is our variance, 111.96 and that's an approximate value because we're rounding there to

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the nearest two decimal places.

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So that's our approximate variance.

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And then of course, to find standard deviation, we just take the square root of variance.

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So standard deviation will be equal to the square root of variance, which in our case is approximately

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the square root of 111.96, which is approximately.

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10.58.

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And so our standard deviation for the sample is approximately 10.58, just over ten and a half.

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Remember and we talked about this before, that the mean is this center of the data, The standard deviation

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is the figure that we'll use as a measure of spread mean is a measure of central tendency.

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Standard deviation is a measure of spread.

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So this figure tells us how much this data is spread out around the mean.

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Remember that if our standard deviation is smaller, it tells us that our data is more tightly clustered

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around the mean, which means all the data gets sucked in closer to the mean.

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And our normal distribution might look something more like this with a taller peak and smaller tails

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where all this data is pushed in closer to this center.

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The mean.

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Whereas if our standard deviation is larger, it tells us that our data is pushed further away from

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the mean.

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It's more spread out.

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This measure of spread is larger, and so our normal distribution might look something a little bit

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more like this, where we have fatter tails and the distribution is shorter, the data is pushed out

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further away from the mean because of that larger standard deviation.

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And we'll talk more later about how we use and interpret mean variance in standard deviation.

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The primary goal here is just to understand how to calculate these three values for a normal distribution

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for both a population and a sample.