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# Introduction to sampling distributions

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.H (LO)
,
UNC‑3.H.1 (EK)

## Video transcript

what we're going to do in this video is talk about the idea of a sampling distribution now just to make things a little bit concrete let's imagine that we have a population of some kind let's say it's a bunch of balls each of them have a number written on it for that population we could calculate parameters so a parameter you could view as a truth about that population we've covered this in other videos so for example you could have the population mean the mean of the numbers written on top of that ball you could have the population standard deviation you could have the proportion of balls that are even whatever these are all population parameters now we know from many other videos that you might not know the population parameter or it might not even be easy to find and so the way that we try to estimate a population parameter is by taking a sample so this right over here is a sample of size n sample of size N and then we can calculate a statistic from that sample based on that sample maybe you know we've picked n balls from there and so from that we can calculate a statistic that is used to estimate this parameter but we know that this is a random sample right over here so every time we take a sample the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter in fact if we were to take a random sample of size n again and then we were to calculate the statistic again we could very well get a different value so these are all going to be estimates of this parameter and so an interesting question is is what is the distribution of the values that I could get for these statistics what is the frequency which which I could get different values for the statistic that is trying to estimate this parameter and that distribution is what a sampling distribution is so let's make this even a little bit more concrete let's imagine where our population I'm going to make this a very simple example let's say our population has three balls in it one two three and they are numbered one two and three and it's very easy to calculate let's say the parameter that we care about right over here is the population mean and that of course is going to be 1 plus 2 plus 3 all of that over 3 which is 6 divided by 3 which is 2 so that is our population parameter but let's say that we wanted to take samples let's say samples of 2 balls at a time and every time we take a ball we'll replace it so each ball we take it is an independent pick and we're going to use those samples of 2 balls at a time in order to estimate the population mean so for example this could be our first sample of size 2 and let's say in that first sample I pick a 1 and let's say I pick a 2 well then I can calculate the sample statistic here in this case would be the sample mean which is used to estimate the population mean and in this for this sample of 2 it's going to be 1.5 then I can do it again and let's say I get a 1 and I get a 3 well now when I calculate the sample mean the the average of 1 and 3 or the mean of 1 and 3 is going to be equal to 2 let's think about all of the different scenarios of samples we can get and what the Associated sample means are going to be and then we can get see the frequency of getting those sample means and so let me draw a little bit of a table here so make a table right over here and let's see these are the numbers that we pick and remember when we pick one ball will record that number then we'll put it back in and then we'll pick another ball so these are going to be independent events and it's going to be with replacement and so let's say we could pick a 1 and then a 1 we could pick a 1 then a 2 a 1 and a 3 we could pick a 2 and then a 1 we could pick a 2 and a 2 a 2 and a 3 we could pick a 3 and a 1 a 3 and a 2 or a 3 and a 3 there's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement and so what is the sample mean in each of these four all of these combinations so for this one the sample mean this one here it is 1.5 here it is - here it is 1.5 here it is - here it is 2.5 here it is - here it is 2.5 and then here it is 3 and so we can now plot the frequencies of these possible sample means that we can get and that plot will be a sampling distribution of the sample means so let's do that so let me make a little chart right over here a little graph right over here so these are the possible possible sample means we can get a 1 we could get a 1.5 we could get a 2 we could get a 2.5 or we can get a 3 and now let's see the frequency of it and I will put that over here and so let's see how many ones out of our nine possibilities we have how many were one well only one of the sample means was 1 and so the relative frequency if we just said the number we could put we could make this line go up 1 or we could just say hey this is going to be one out of the nine possibilities and so let me just make that I'll call this right over here this is 1/9 now what about 1.5 well let's see there's one two of these possibilities out of nine so 1.5 it would look like this this right over here is 2 over 9 and now what about - well we can see there's one two three so three out of the nine possibilities we got a two so we could say this is 2 or we could say this is 3/9 which is the same thing of course as one-third so this right over here is 3 over 9 and then what about 2.5 well there's - 2.5 so 2 out of the 9 times another way you could interpret this is you have 8 when you take a random sample with replacement of two balls you have a 2/9 chance of having a sample mean of 2 point five and then last but not least all right over here there's one scenario out of the nine where you get to three so 1/9 and so this right over here this is the sampling distribution sampling distribution for the sample sample mean for for N equals two or for sample size sample size of two
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