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So we've said that measures of central tendency attempt to measure or indicate some kind of center point

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of the data set.

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But now we want to talk about measurements of dispersion or measurements of spread, which really at

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their core, attempt to quantify how much the data is spread out around the mean.

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So let's return first to this idea of a population which we indicate with Capital N or a sample, which

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we indicate with lowercase n.

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If we are looking at an entire population, for instance, maybe all of the students in our school,

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if we're thinking about students at the school, 100% of those students make up the population.

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And we would indicate that entire student population with capital N.

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If we want to take a sample of students in the school in an attempt to find a smaller representative

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group of that larger population, we would take some subsection, some sample of that larger population

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and we would indicate that sample with lowercase n.

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So if there are 1000 students in the school, then capital N is equal to 1000.

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If we're taking a sample of 100 students, then we would say lowercase n is equal to 100.

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And of course, as we already know, the mean for both the population and the sample we can find using

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the same formula.

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Remember that these two formulas are identical.

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The only difference is that when we indicate a population mean, we usually do so with this Greek letter

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MU, and when we indicate the mean for a sample, we usually do so with this x bar notation.

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And as we're summing up all of the data points in the population, we indicate that population here

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with Capital N and we divide by capital N the number of subjects in the population.

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Whereas with the sample mean formula, of course we're using lowercase end to indicate the number of

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subjects in the sample.

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But otherwise these formulas are going to calculate the same thing just one for the population and one

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for the sample.

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Now, when it comes to measurements of dispersion or measurements of spread, we're talking about how

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much the data is spread out around the mean.

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When we looked at measures of central tendency, we talked about mean, we talked about median, we

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talked about mode.

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But here we're really focusing on mean.

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And when we're talking about measures of spread, we're interested in how the data is spread out around

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this mean value.

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Is it very tightly clustered around the mean where all the data sits really, really close to that mean

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value, or is the data really spread out far from the mean?

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Does it have a larger range?

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Is a significant amount of the data far away from this mean value.

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That's what we're talking about when we say measures of dispersion or measurements of spread, how much

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is the data spread out?

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How much is the data dispersed from the mean?

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And when it comes to those measurements of dispersion or measurements of spread, we look at two things

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variance and standard deviation, both of which are very closely related to one another.

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Let's start by looking at this idea of variance, and again, we can calculate the variance for the

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population Capital N or for the sample lowercase n, and with both variance and standard deviation,

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the smaller the value we find, then of course the smaller the variance or the smaller the standard

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deviation and the more tightly clustered the data is around the mean.

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If we find a larger value for variance or a larger value for standard deviation, which we'll look at

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in a second, that means that the data is more spread out, that it is farther away from the mean.

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Now, when it comes to calculating variance, if we're talking about population variance, we indicate

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that with this Greek letter sigma and when we talk about population variance, we call that value sigma

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squared.

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In the same way, when we talk about sample variance, we talk about that as this value F squared.

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So we use sigma squared for population variance, SX squared for sample variance.

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And you'll notice that just like the formulas with the mean, these two formulas are essentially identical,

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except that here in the population formula we are using population mean mu you'll see that value here

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MU and with the sample we're using the sample mean value x bar that we already calculated up here when

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we found sample mean.

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And of course we're using capital N in the population formula lowercase n in the sample formula.

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Now these variance formulas look a little complicated, but really all they're calculating is the mean

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of the squared deviations.

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So what do we mean by that?

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Well, let's look at this population variance formula here.

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We already know from the formula we looked at for mean that x abi is just one data point in our population,

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one value, one subject in our population.

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And we know that mu here is population mean.

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So when we take x abi minus mu, this value right here is giving us the distance between this one data.

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And the mean of the data set.

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So if we found that the mean of the population was four and this particular data point is ten, we would

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take ten minus four and we'd get a value of six here, which just indicates that the particular data

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point of ten is six units away or a distance of six away from this population mean we call that the

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deviation.

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How much does ten deviate from the mean of four?

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Well, it deviates by six units or this idea of distance of six.

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So this is just the deviation of each particular data point away from the mean.

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And of course, once we find that deviation, then we square it.

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So this whole value here, quantity x sub minus mu squared, we say that that is the squared deviation.

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And then this notation here just tells us to add up all of the squared deviations, and then we're dividing

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by the number of subjects in the population capital N but adding up all of the squared deviations and

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then dividing by the number of squared deviations.

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That's just like what we did when we took the mean.

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We added up all the data points and then we divide it by n here we're adding up all the squared deviations

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and dividing by N.

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So this whole part of the formula here, this sigma notation and the divide by capital N is just taking

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the mean of the squared deviations.

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So this variance formula is really just calculating the mean of the squared deviations.

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And of course that's the same for sample variance.

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And so you can start to get an idea that what we're really doing here is we're kind of trying to find

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a mean distance away from the mean itself.

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In other words, in simple terms, on average, how far is each data point away from the mean away from

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the center of the data set?

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00:06:45,250 --> 00:06:50,860
Because if we can find an average of how far away each data point is from that center point, then we

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can start to get an idea that, oh, the data is really spread out or that variance value is small,

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the data is really close together.

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The only thing about this concept of variance that tends to throw people off is this squared exponent

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here.

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The fact that we're squaring the deviations, there's a couple of reasons why we square this deviation

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value.

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First, if we get a value x sub I that is less than the mean.

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So thinking about our example before we said that maybe the mean of the population is four.

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Well, if this particular data point in the population is two, we're going to take two minus four,

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we're going to get a negative two.

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Our deviation is now negative and things can start to get a little strange.

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If we're working with negative values and trying to find a mean of both negative and positive values.

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So one reason that we square this deviation is to force all of these deviations to be positive.

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If we get a deviation of negative two and we square it, it becomes a positive four.

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And so then when we start adding up all of these squared deviations, we're going to be adding only

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positive values.

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It makes this calculation a lot cleaner.

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So we square this value to get everything to be positive, which makes our calculation easier.

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And the other reason we do it is because squaring the deviations magnifies the larger deviations and

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forces them to have a bigger effect on the variance than smaller deviations.

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So you can imagine here, let's say that we find a deviation this x sub minus mu value.

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If we find a deviation of ten and we square it.

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Now the squared deviation for that particular data point is 100.

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Whereas if we find a deviation of two and we square it, our squared deviation is four.

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So we're talking about the difference in deviations being two versus ten, but the difference in squared

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deviations being four versus 100.

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Now all of a sudden that difference is much, much larger.

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So it's really magnifying those values that are further from the mean.

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And that's going to help give us a clear picture of how varied, how dispersed the data really is away

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from the mean.

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So that's the variance concept.

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And then our other measure of dispersion is what's called standard deviation.

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And luckily standard deviation is very, very easy to calculate once we calculate variance and once

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we understand what variance is, because standard deviation is just the square root of variance.

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So we use this sigma letter, Greek letter sigma to indicate standard deviation.

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So notice that variance for the population with sigma squared, but standard deviation for the population

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is just sigma.

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It is the square root of sigma squared.

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So to find standard deviation, we just take the square root of sigma squared, which of course is variance.

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So we could also replace this sigma squared value underneath the square root with the entire right hand

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side of this population variance formula that would give us another formula for standard deviation.

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And then same thing over here.

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For sample standard deviation, we indicate that with SE, which of course is just the square root of

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x squared, and we know that SE squared is sample variance.

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So.

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Once we calculate population variance defined population standard deviation, we just take the square

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root of population variance.

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Or once we've calculated sample variance to find sample standard deviation, we just take the square

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root of sample variance and that'll give us sample standard deviation.

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Now some people wonder why we would ever bother using standard deviation.

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Why would we bother taking the square root to get a standard deviation value when this standard deviation

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value feels so similar to variance?

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And we've already calculated variance like what's the point of standard deviation?

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Well, the reason that we like to find standard deviation value and in fact we usually work with standard

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deviation instead of variance.

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So we'll talk much, much more about mean and standard deviation than we ever will about mean and variance

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and going forward will usually just use variance as a step on our way to finding standard deviation,

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which we're more interested in.

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The reason we're more interested in standard deviation is because the units of standard deviation will

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match the units of the mean and the units of our data set.

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So for instance, if you think about maybe a data set where all the data is given in terms of centimeters,

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so let's say that we have some sample of people's heights and that data is given to us in terms of centimeters.

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Well, when we go to find variance, the units on this x sub value are going to be centimeters.

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The units on X bar, the mean are also going to be centimeters, because all we did up here to find

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the mean was add up a bunch of centimeter values and then divide by some whole number.

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N And so the units of X bar are going to be in centimeters.

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We're going to have a mean in terms of centimeters.

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And so then when we calculate variance, the units of x sub by minus x bar are also going to be centimeters.

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But then when we square that value, we're going to end up with a centimeters squared.

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We're going to add up all those centimeter squared values, and we're going to have a centimeter squared

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unit value in this numerator.

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And then we'll divide by a whole number and we'll get a value for variance as squared in terms of centimeter

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squared.

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So if our units for the mean and for the data itself are in centimeters, then variance by definition

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is going to be in centimeter squared.

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But of course, as you can imagine, when we take the square root of this, when we take the square

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root of variance to get standard deviation, the square root of centimeter squared will just be centimeters

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again.

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And so the units of standard deviation are going to be back in terms of centimeters.

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And therefore the units of standard deviation and mean are always going to match one another, which

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is really nice.

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It's going to make these two values easier to interpret and compare and match up.

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And so that's a big part of the reason and one easy to understand reason for why we like to work with

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standard deviation, with the mean instead of variance with the mean.

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So all that being said, now that we understand mean variance, standard deviation and how we're calculating

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these, let's just do one quick example to make our calculation easy to do by hand.

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Let's pick a really super simple data set and say that we have some data set where the values are.

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We'll just pick something really easy one, two, three, four and five.

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Let's say that this is sample data.

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We've taken a sample of our population and so we can see that we have five subjects in the sample data

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and so we can say lowercase n is equal to five.

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These are our five sampled data points.

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And so if we take the mean, we should be able to see right away that the mean is three because we have

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this balancing point for the data here of three, but we could also use this formula to calculate sample

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mean by adding up these values.

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If we add up one, two, three, four and five, we get 15.

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And if we divide by N equals five, we get 15 over five and we find three.

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So we can say that x bar, the mean of the sample is equal to three.

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So we have our sample mean.

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00:14:17,120 --> 00:14:23,780
Now we want to talk about how to find the variance of that sample just so we can see this formula in

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00:14:23,780 --> 00:14:24,440
action.

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So again, we're going to start by looking at the deviation of each data point in the sample.

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So we're going to take this x sub value for our first data point.

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00:14:34,700 --> 00:14:42,350
So that's one we'll take one minus three, that's x sub, B minus x bar, and that's our first deviation

195
00:14:42,350 --> 00:14:43,280
for the sample.

196
00:14:43,280 --> 00:14:48,980
So if we want to calculate here sample variance, we'll say x squared will do it down here.

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00:14:48,980 --> 00:14:55,490
And our first deviation we said was one minus three, so one minus three quantity squared and we're

198
00:14:55,490 --> 00:14:58,190
just going to add up all of the squared deviations.

199
00:14:58,190 --> 00:14:59,360
So next we have.

200
00:14:59,470 --> 00:15:01,380
You minus three quantity squared.

201
00:15:01,390 --> 00:15:04,130
So two minus three squared.

202
00:15:04,270 --> 00:15:11,440
Then we take our third data point and we get three minus three quantity squared and then our fourth

203
00:15:11,440 --> 00:15:12,340
data point.

204
00:15:13,460 --> 00:15:18,550
And then our last data point five minus three quantity squared.

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00:15:18,560 --> 00:15:23,300
So this is the sum of all of the squared deviations.

206
00:15:23,300 --> 00:15:28,700
And then we're going to divide that by MN minus one for sample variance.

207
00:15:28,700 --> 00:15:30,870
So we already know lowercase n is five.

208
00:15:30,890 --> 00:15:36,340
There are five data points in the sample, so n minus one is five minus one or four.

209
00:15:36,350 --> 00:15:38,510
So we get four in the denominator.

210
00:15:38,540 --> 00:15:44,120
Now, before we go forward, let's talk really briefly about this n minus one value, because you may

211
00:15:44,120 --> 00:15:50,120
be tempted to think that this denominator should just be lowercase n in the same way that the denominator

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00:15:50,120 --> 00:15:55,040
for population variance is capital N and not capital N minus one.

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00:15:55,040 --> 00:15:58,820
So the question is why is this lowercase n minus one?

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00:15:59,090 --> 00:16:05,510
Well, the reason is because, of course, if we have our entire population, if we know all the data

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00:16:05,510 --> 00:16:11,550
from the population, then we can calculate a perfectly accurate population, mean mu.

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00:16:11,570 --> 00:16:17,300
And we can go through and find a perfectly accurate population variance and population standard deviation.

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00:16:17,300 --> 00:16:23,330
But when we're taking a sample, of course, remember that taking a sample automatically introduces

218
00:16:23,330 --> 00:16:30,950
some variability into our calculations because of course we might take one sample as we did here and

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00:16:30,950 --> 00:16:38,540
find a mean X bar equals three, but we might take a different sample of five subjects and find a different

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00:16:38,540 --> 00:16:38,870
sample.

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Mean for that sample.

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Maybe that sample mean we find is x bar equals five.

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We could find another sample, mean x bar equals four.

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So the mean that we find the sample mean we find depends on the exact sample that we pull from our population.

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And the sample mean might change each time depending on the sample we pull.

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So we've got some variability here in this sample mean.

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So when we jump down here then to calculating sample variance, what we're doing here when we find these

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deviations is we're taking all of the values in our sample like up here, one, two, three, four and

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five, four x sub I, and we're finding the distance between those values and this sample mean x bar

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equals three.

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But of course by the nature of the sample mean we calculated x bar.

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We have already in a way minimized the distance, minimized the deviation of all of these sample values

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from x bar because of course this x bar value we calculated this sample mean that we calculated is by

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definition the single value we can find that is as close as possible to all of these values from our

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sample.

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It is the balancing point of all the values in our sample.

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And so we're already in a way minimizing these deviations.

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So it's kind of this idea that we are going to potentially underestimate deviation.

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And so the reason that we divide by n minus one is to correct at least a little bit for that bias.

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In other words, dividing by N minus one helps us undo a little bit of the bias that we introduced when

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we minimized the deviation in our sample by calculating this x bar value and then finding the deviation

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of each of these data points in the sample from that sample mean that we calculated.

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So that's why when we find population variance, we divide just by population capital N But when we

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find sample variance because of the nature of sampling and the bias and variability that we introduce,

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we divide instead by lowercase n minus one.

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And so in this case, this example we're working with, there were five subjects in the sample, so

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we divide by five minus one or four.

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So then all we have to do is calculate.

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Here we get one minus three is a negative two, and when we square that we get four.

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Two minus three is a negative one.

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When we square it, we get one.

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Here, we have zero, here we have four.

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Minus three is one squared is positive one and five minus three is two, quantity squared is positive

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four.

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So we get four and then we divide by four.

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And of course the value here then is ten over four or five over two.

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Or we could also write that as 2.5.

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So our sample variance then if this is our sample, our sample variance is 2.5.

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And then of course if we wanted to find standard deviation, all we have to do is take the square root

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of that value.

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And so we would say then that if we put up here sample variance is five halves, then sample standard

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deviation is the square root of five halves, the square root of five halves, and that value is approximately

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equal to.

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1.58.

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And this is, of course, like we said, 2.5.

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So hopefully that gives you an idea of what we're doing when we calculate variance or standard deviation

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for a population or a sample, both of which variance in standard deviation, both of which are measurements

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of dispersion or measurements of spread that give us an idea of how much the data is spread out around

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the mean, whether the data is spread out far from the mean, there's a wide range of the data away

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from the mean or whether that data is super tightly clustered around the mean.