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We've already talked about the idea of a population and a sample where a sample is a subset of the population,

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and we're trying to take a sample from the population that is representative of that population.

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But now we want to talk about this idea of comparing information about the sample to information about

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the population, for instance, the population mean versus the sample mean.

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So the first thing we want to say is that when we talk about information that's related to the population,

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we call those the parameters of the population.

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So the population has the parameters of size mean and standard deviation given by capital nn mu and

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sigma respectively.

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And in comparison, if we talk about these same characteristics or the same set of information that

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relates to the sample, we call those values statistics.

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So characteristics of the sample are statistics, characteristics of the population are parameters.

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And we already looked at these values before the size of the population is given by capital N, whereas

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the size of the sample is given by lowercase M, we indicate the mean of the population and sample with

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MU and x bar respectively, and the standard deviations of population and sample are given by Sigma

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and S, And the whole idea behind the field of statistics is that we are sampling, we've talked about

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sampling already.

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We are sampling from the population.

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We are collecting statistics from that sample.

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For instance, we're looking at the mean and the standard deviation of the sample.

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And our goal, our hope is to be able to use these statistics to make inferences about the corresponding

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parameters from the population.

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In other words, we want to be able to look at the sample mean and use the sample mean to get what we

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hope will be an accurate estimate of the population.

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Mean because after all, it's these population parameters that we're really interested in.

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We want to know maybe the mean and standard deviation for the entire population.

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And if we could actually calculate that value specifically by collecting data or surveying the entire

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population, we certainly would.

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We're only sampling because it's very difficult or impossible to collect data from the entire population.

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So the best we can do is sample gather statistics and then use these statistics to make inferences about

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the corresponding parameters.

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The problem is that if we just collect one sample from the population and we compute a mean and a standard

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deviation for that single sample, the values we find for sample mean and sample standard deviation

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may or may not be good estimates of the mean and standard deviation of the population.

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In other words, if the sample is very representative of the population, if it does a good job representing

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the population, then our statistics might be good estimates of the parameters, but we might just happen

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to pick a sample that is not very representative of the population or does a bad job representing the

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population, in which case our sample statistics are going to do a bad job estimating their corresponding

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population parameters.

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Now to try to prevent this problem where the statistics we find are bad estimates of the corresponding

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parameters, we can think about the idea of taking many samples instead of just one sample.

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You can imagine that if we take just one sample, that sample will have its own mean and maybe we call

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that sample mean x sub one like this because it's the sample mean for the first sample and the standard

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deviation of that first sample will call s sub one, but then we might take another sample.

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And that sample mean we might write like this X to with that standard deviation for that second sample

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being as two.

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And maybe we take a third sample and we compute the mean and standard deviation for the third sample.

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And obviously we could continue on and eventually if we keep sampling over and over and over collecting

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many, many samples, we end up with a whole set of sample means x1x2x3x4x5 and on and on through many

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samples.

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So we have this whole set of sample means and it turns out, and this might feel somewhat intuitive,

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that as we take many, many sample means more of those sample means will turn out to be closer to the

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population mean mu and fewer of those sample means will turn out to be further away from the population.

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Mean mu and in fact that set of sample means will form its own probability distribution around the population.

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Mean mu, and that probability distribution of sample means will almost always be a normal distribution.

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This normal distribution has its own mean and if you think about it, we could call that mean the mean

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of all the sample means right.

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This is the probability distribution of the sample means the means of all those different samples that

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we took and we could calculate a.

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I mean, of all of these sample means.

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So this probability distribution has its mean right at the center here as the mean of all these sample

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means and that mean of sample means.

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If we put it in maybe right here we call mu sub x bar because x bar is how we indicate a sample mean.

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And so this is the mean of the sample means.

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And it turns out that this probability distribution is centered at or has its own mean the mean of the

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sample means and that this mean of sample means will be equal to.

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And this is the amazing part the population mean and this is in fact the conclusion of the central limit

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theorem which is an amazing conclusion.

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It tells us that if we continue to take samples and we find a sample mean for each one of those samples

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and so we get the sample means x1x2x3.

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If we plot those sample means along a number line, then it turns out that more of these sample means

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are going to cluster around the population mean.

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So we might calculate the first sample mean and find that its value is let's say right here.

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And then we calculate a second sample mean and we find out that its value is here.

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We calculate a third sample mean and we find out that its value is here.

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And as we continue calculating more sample means taking more samples and finding the mean of each sample.

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We see that those means start to cluster more heavily around this one value.

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Here.

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Of course, there's some all over this distribution, but most of them are clustered in the center here

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at this value.

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And then fewer of them out to the right here and fewer of them out to the left here.

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But most of them are clustered around the mean of the sample means, which is going to be equal to the

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population mean.

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And so this is amazing here because what we're saying is that we have a population and we don't know

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the population mean we have no idea what it is.

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But with enough sampling, we can create this distribution here.

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And this distribution is almost always going to be a normal curve, a technically normal curve, that

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bell shaped curve that is symmetrical with its mean.

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Here at the center, we call this distribution the sampling distribution of the sample mean or SDS.

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M Which makes sense because we're creating a distribution by sampling, and this is the distribution

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of all of the sample means.

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So it's the sampling distribution of the sample mean.

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And the central limit theorem tells us that this distribution will be a normal curve.

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So what this is really allowing us to do is turn a population that is non normal, that is not normal

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into a distribution that follows a normal curve here.

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And once we have this normal distribution, we can convert it into the standard, normal distribution

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and use Z scores to answer all kinds of probability questions related to this distribution and in that

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way make educated guesses about how accurate our statistics are as estimates of their corresponding

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population parameters.

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Now, a couple of things we should say before we just move on from the idea that we have a normal curve

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here for the sampling distribution of the sample mean.

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The first is that if the original population is normally distributed, then the sampling distribution

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of the sample mean will also be normally distributed regardless of the sample size that we use.

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In other words, if the population is normal, then we don't really have to worry about the size n of

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our sample.

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We know that the sampling distribution of the sample mean will also be a normal distribution.

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It'll be a normal curve.

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But if the population is not normally distributed, or if we just don't know whether or not it's normally

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distributed, then the sampling distribution of the sample mean is only guaranteed to be normal if we

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use a sample size of at least 30.

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So n has to be greater than or equal to 30 in order to ensure that this curve is a normal curve.

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So if at all possible, it's good practice to use a sample size of 30 or greater to ensure that we're

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working with the normal curve here.

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If we don't know whether or not the population itself is normal, which will often be the case, we

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often will not know for sure whether or not the population is normally distributed.

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So now that we have a sampling distribution of the sample mean, we recognize that this is a normal

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curve.

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And so we can talk about the mean and standard deviation of this specific distribution.

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Specifically, we can say that the mean of this distribution, which we said we can think about as the

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mean of the sample means is equal to the population mean.

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So that's our first point there.

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Then the variance of this distribution is going to be equal to sample variance divided by sample size.

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So the variance of the sampling distribution of the sample mean will be equal to the sample variance

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of the single sample that we took.

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Divided by the sample size for that single sample.

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And therefore, remember, that standard deviation is always the square root of variance.

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So as you might expect, if we take the square root of SE squared divided by MN, the square root of

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SE squared is just SW, and the square root of N is the square root of MN.

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So the standard deviation of the sampling distribution of the sample mean is se divided by square root

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of MN or the standard deviation of our single sample, divided by the square root of the sample size.

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And we write that standard deviation as the standard deviation with x bar here, because it's the standard

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deviation of the sampling distribution of sample means.

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And specifically, this value is important enough that we give it a special name.

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We call it the standard error.

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In other words, the standard error is another name for the standard deviation of sample means or the

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standard deviation of the sampling.

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Distribution of the sample means.

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Now, of course, just like with any other standard deviation value we've talked about before, when

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this standard deviation is larger, when the standard error is larger, it tells us that the sample

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means in the sampling distribution of the sample mean are more spread out away from the mean, which

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means that any one particular sample mean is less likely to be an accurate representation of the true

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population mean.

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Whereas when standard error is smaller, when this standard deviation is smaller, it means that all

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these sample means are more tightly clustered around this mean here, which means that any one sample

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mean that we take is more likely to be an accurate representation of the actual population.

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Mean.

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In other words, standard error gives us an idea of how likely it is that any given sample mean is an

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accurate representation of the true population mean.

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So ideally, what we want is a small standard error because the smaller we can get the value of standard

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error, the more likely it is that our sample mean for our one single sample that we took from the population,

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the more likely it is that that sample mean is a good representation of the population mean, which

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is ultimately what we want because we want to know population mean.

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So we're hoping that when we take a sample the mean of that sample will be close to the population mean

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and the smaller the value of standard error, the more likely that is to be true, the more likely it

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is that the mean of the single sample we took is close to the population mean.

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So then the idea becomes how do we decrease the value of standard error?

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How do we get a really small standard error?

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Well, when you have a fraction like this one, we have a sample standard deviation divided by square

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root of sample size.

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When you have a fraction, there are two ways to decrease the value of the entire fraction.

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We can either decrease the value of the numerator, which in this case would mean decreasing sample

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standard deviation and or we can increase the size of the denominator, which in this case means increasing

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the size of the sample.

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Now, there's not a whole lot we can do about the sample standard deviation, because the value of sample

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standard deviation is just a result of the data that we collect from the sample.

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It just turns out to be whatever value it is based on the sample that we get.

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What we can do is increase the sample size and that should make sense to us at an intuitive level.

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The larger our sample, the more accurate our statistics are going to be as representations of the parameters.

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If our total population, let's say, is 1000 people and we take a sample of just ten people, intuitively

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we know that that sample could turn out to be really inaccurate.

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It may not be representative of the population at all, but if we have a population of 1000 people and

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our sample is, let's say 700 people, we drastically increase the sample size, being able to collect

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information from 700 of the 1000 total people, a 700 person sample of a 1000 person population.

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We're probably going to get a pretty accurate idea of what that population looks like with a sample

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that big.

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So as we increase the sample size, we probably got a more accurate sample mean and therefore a smaller

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sample standard deviation.

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And so our standard error is smaller and it's more likely that our sample mean is an accurate representation

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of the population mean.

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So the conclusion there is that taking a larger sample is going to help improve the accuracy of the

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sample mean as a reflection of the population mean.

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But of course taking a larger sample might mean that we have to spend more time or money on our sampling

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process.

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It might be more difficult to take a larger sample.

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So in the real world, we're always balancing these things.

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Maybe we want to take as big of a sample as possible, but we're limited by our manpower or money,

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our resources, and so the sample we take can only be so big.

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Now, the other thing we want to say about these formulas for variance and standard deviation is that

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depending on the size.

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With our population and the size of our sample, we may need to apply what's called the finite population

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correction factor, which looks like this.

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So the finite population correction factor.

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FPC for short, we'll talk about when we need to apply it, but when we do need to apply it, the formula

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for variance changes to this formula.

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Here we keep the SE squared over n, but we have to multiply by capital n minus lowercase n.

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In other words, population size minus sample size, divided by population size minus one.

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And then the formula for standard error or standard deviation of the sample means is the same as divided

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by square root.

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N.

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But then we have to multiply that by square root of this population size minus sample size, divided

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by population, size minus one.

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Now, when do we have to apply the finite population correction factor?

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Well, we have to do it when we are sampling without replacement and or when we're sampling from more

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than 5% of a finite population.

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So here's what that means.

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Remember previously we talked about taking a simple random sample.

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Well, whenever we're sampling, ideally we're sampling with replacement, which means that in theory,

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the same person can be picked multiple times for our sample.

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So, for instance, to take a simple example, let's say our population is all the people who live in

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our neighborhood, which maybe is 1500 people.

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If we're sampling with replacement, it means that we randomly choose one person in the neighborhood

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and we record them as part of our sample, but then we sort of throw them back into the pot and we pick

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another person to be included in our sample.

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We throw them back in the pot and then we pick a third person, throw them back in the pot, pick a

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fourth person, until eventually we have our full sample.

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What that means is that because every time we choose to include someone in our sample and then we sort

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of put them back in the population, in theory, we could put that same person multiple times for our

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sample.

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Two times, three times, four times, really, because it's random chance we could pick them every

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time for our sample.

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That's the idea of sampling with replacement, The same subject object person, whatever it is in the

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population can be chosen multiple times for the sample, but sometimes we won't be able to sample with

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replacement.

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Maybe we're working for a political campaign and we're asking people as they leave their voting location

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about how they voted and we're collecting that information for our sample.

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Well, obviously that person's only going to leave the voting location one time.

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We're going to ask them our questions on the way out the door and then we're never going to see them

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again.

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In that case, we're sampling without replacement.

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It's not possible for us to pick the same person for our sample multiple times.

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And so we would be sampling without replacement.

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If that's how we're collecting our sample, then we would need to make sure that we include this finite

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population correction factor in the formulas for variance and standard deviation.

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For this sampling distribution a sample means.

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And then the other sampling condition when we apply the FPC is when we are sampling for more than 5%

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of essentially a finite population.

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So again, going back to our neighborhood example, when our population is the 500 people who live in

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our neighborhood, 5% of 4500 is 75 people.

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So if we are taking a sample that is 75 people or larger, then we would want to apply this finite population

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correction factor.

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In that case as well.

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We only apply the correction factor in these conditions because outside of these conditions, it turns

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out that this finite population correction factor, this value here, turns out to be very, very close

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to one.

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And when its value is close to one, obviously it's not going to affect the value of variance or standard

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deviation.

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And so we don't have to include it.

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The value of the correction factor drifts away from the value one under these conditions and therefore

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does start to have an effect on variance in standard deviation, which is why we have to include it

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under these particular conditions.

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So that's the idea behind the central limit theorem.

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And it's critical because what it allows us to do is take one sample from our population and with this

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justification of the central limit theorem telling us that the sampling distribution of sample means

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is normal, a normally distributed curve around the mean of sample means, which is equal to the population,

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mean What we'll be able to do with that conclusion a little later on is from the one sample.

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Calculate the sample mean and sample standard deviation for that one sample.

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And then because we have this normal curve and we know how to use Z scores to answer probability questions

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about data that's normally distributed, we'll be able to make a statement about how likely it is we'll

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be able to give a specific percentage about the likelihood that this sample mean the mean of our one

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sample that we took falls within.

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Some interval around the population mean.

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In other words, we might find a sample mean let's say our sample mean is 14 and then we'll learn later

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on about how to calculate an interval around this sample mean.

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But let's say our interval is one and one half units to each side of this sample mean, which takes

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us to 12.5 on the left and to 15.5 on the right.

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And so we end up with this interval of 12.5 to 15.5 and we might be able to make a statement along the

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lines of we are 95% confident that the population mean falls somewhere in the interval 12.5 to 15.5,

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which of course is a very powerful kind of conclusion.

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We can say how confident we are or how likely it is that the actual real population mean will fall within

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some particular interval, even though we have no idea what the actual population mean is.

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And all we did was take one single sample from that population and calculate a sample mean of 14.

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But using just that one data point, just that one statistic, one sample mean because of the power

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of and the conclusion of the central limit theorem as it relates to the sampling distribution of the

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sample mean we can make a statement about our confidence level that the population mean falls within

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some particular interval around the sample mean that we calculated.