One of the things that I get asked about most often with respect to surveys and survey sampling, especially by people who analyze surveys, is about weights. And as long as we're talking about stratified random sampling and the idea of weights that are used to combine results across strata, I thought it might be useful to also talk about how that can be converted and considered in terms of weights at an element level. And the reason this comes up is because that kind of weighting at the element level to combine our results across strata is how software handles it. And we should be prepared for that. The software is not going to do what we've just described. It won't even compute the variances the way we would describe. But we want to begin to get something of an introduction to the ideas behind weighting at the element level as we're talking about stratified sampling. So let's consider then what we mentioned before, that the weighting, the sample is necessary. What we're going to do is compare or combine across groups to get back to conclusions about the total population. Recall that some of our distributions or allocations, we had cases where we allocated much more of the sample to a group than its proportionate size in the population. And we talked about how well the weights, you know those W sub Hs, would correct for that when we put things back together. And that would be great if that's the way that the software expected the data to come, but it doesn't. And it's because we can convert that into another form, into a second way of doing the weighting, that we need to consider this a little bit. So in practice, when we're using statistical software, it's done using only one of the two ways. Not the population weighting, and the W sub Hs, but actually an element level weighting. And you see in the lower left a formula that deals with the weighted mean. We're going to come back to the weighted mean as we look at other kinds of weighting problems that come up in Unit 6. So one weighting method that we've been talking about, just to be more specific, is we're going to weight the stratumus that's by the size of the strata. So our y bar is the sum of the y bar sub H, the means in each of our strata, combined with the weight factor, capital W sub h, the relative sizes of each of the strata. Unfortunately as I say this is not done in the software. If the software were to do this kind of thing, it would need to know not only the strata that we've got. That would be natural and we're going to specify that when we do variance estimation what stratum each of the elements is in. But also the W sub h. We're going to have to give that in a separate string. And if we have a lot of strata, it could be very tedious to do. So the software has never gone in that kind of direction. What they've done is to weight by elements. Because the weighting by elements not only deals with the stratified sampling case, but a number of other cases where we have disproportion allocations, but generated in a different way. So there is a way to get the same mean using element weights, rather than stratum weights. And that can be done if what we do is use a weight, and I'm just going to give you the result, if we use a weight at the element level, that is capital N sub H/n sub H. That is the population size divided by the sample size. That is the inverse of the sampling fraction. It's the inverse of the sampling rate within the strata. If we use that weight, and weigh each element now, not each stratum mean, but each element within each strata by that weighting factor. And we multiply each value by that weight and add it up and divide by the sum of those weights, we get exactly the same mean. That is an issue that we're going to need to consider in more detail and understand why these sampling rates, the inverse of sampling rates would give us the actual estimate that's obtained by using the stratum weights. We'll come back to this in Unit 6 when we talk about weighting as one of our three main topics in Unit 6. But at this point, it's just worth keeping in mind. Stratified random sampling, we can compute our estimates by weighting at the stratum level, computing everything within the strata, and then multiplying by the weight when we're computing the mean. Or by the square of the weight, when computing the variances. But we can also do that same kind of calculation weighting, as we've seen here, by the inverse of the sampling rate within each of the strata. But before we get to talking about weighting in Unit 6, there's another sample design that we're going about in Unit 5. And that's one that's called systematic sampling. Sometimes it's referred to in literature as pseudo random sampling. It's a simplified kind of sampling. And with simplified kind of sampling, it helps us when we're dealing with clerically complicated kinds of sampling procedures to have something that is very easy to do, a much more straightforward way of dealing with this. And so we're going to talk there about such factors as selection and list order, and intervals, and estimating standard errors for systematic samples. Just to give us another tool in our kit of randomized selection and simple random sampling, cluster sampling to save money, and stratified random sampling to make ourselves more efficient and address such issues as domain estimation. So please join me in Unit 5 as we talk about systematic sampling and its applications. Thank you.