Welcome, once again to our Coursera course on sampling methods in the context of data collection and analysis. In this unit, we're starting a new unit, we're going to talk about randomization as a sampling technique. What I'm calling near randomization, only randomization. We're just going to look at this as the primary selection device in our process. There is only one way an element in the frame, called our frames, can get into the sample, and that's by individual random selection. It is typical to call this kind of thing simple rather than mere random sampling. Simple referring to just a randomization. But I hesitate to use the term, even though it's more commonly used because this is less simple than it might first appear. There's a part of simple random sampling that depends on a choice of duplicating elements in the sample that bothers statisticians because of the finite bases the population. And it's that application of mere randomization to finite population that will cause us some problems. But that goes beyond where we are for right now. So let's introduce SRS, as this is sometimes called, simple random sampling. And we're going to talk a little bit about frames and random variables. We're going to talk a little bit about selecting a sample and some definitions. And then the practical use of simple random samples as we go along. You recall that we've already done a sample random sample. We had a rather large table of random numbers, when we were talking about randomization, and what random numbers look like. Well, here's that table of random numbers. I know it's impossible to read, but you can maybe pull this out, zoom it in and look at the properties. All of these numbers, in this case, 4,500 numbers that could be used to draw our sample. And then we apply this set of random numbers to our frame. Our frame that consists of persons who happen to be faculty. So we're sampling persons who are faculty, professors, associate professors, assistant professors at a university. And we see that there's a little different layout than before, because we can see their sequence number, and an ID that is somewhat sequential, but with gaps. And a little bit of other information, auxiliary data. Auxiliary data about their rank, about their division where they happen to be appointed, their sex. But we're going to ignore that information. We're just going to treat this as a list of individuals and apply our random selection techniques to that list of individuals. The frame has more elements than can fit on one sheet, so here's another sheet. There's at least 300 of them in this particular frame. As a matter of fact, there's a total of 370 elements, that's our capital N, our population size, as far as we're concerned. There could be more faculty, but we're just limiting it to these 370. And here was the simple random sample that we did in our last unit. We didn't call it that, but here is what we did. So let's sort of review the steps. We identified random numbers from the list that we could then link up to the ID numbers, or the sequence numbers for the subjects. So we were looking for random numbers that went from 001 to 370, and we're looking for in our case from that final population of 370 elements, 20 such numbers that were distinct, different. And we're using random numbers from a table and matching them up to unique numbers assigned to each frame element. And for each selection after the first, we went back and checked to be sure. Well, maybe we didn't do this formally, we probably did it informally and very fast. We checked to make sure that when we had selected the next unit, it wasn't one that had already been chosen. Because if it was, we went back and rejected that most recent selection, which was a duplicate of one that was already there, and then generated another random number, and made another selection. That is we made sure that we had 20 distinct, 20 different elements from the population, our sample. And we just continued that process, again and again doing the draws, until we got to our sample size of lowercase n. 20 distinct frame elements selected. So this we labeled without replacement selection. A unique sample of elements. With that kind of an approach, where we don't allow duplication, you can see the little key there that says don't duplicate. Every element of the population has the same probability of selection. Now we're going to use a term here throughout the course to refer to that process and call it epsem, E-P-S-E-M. You see that highlighted in red. It's an abbreviation. An abbreviation for equal, E for equal, probability, P for probability, S-E for selection, method, m for method. Equal probability selection method. Every element in the population has the same chance of getting into the sample as any other. And further, every combination of size lowercase n 20, in our particular case, has the same probability of selection, the groups themselves, all those unique sets of 20 different elements, and there are lots of them, have the same chance of selection. All right, that's what we did before. That was our simple random sampling process. There's other ways to do this and get a simple random sample in the end the same way. So again, we're interested in a sample of size lower case n from capital N, 20 from 370. And here's what we're going to do. We're going to change the process. We're going to get our random numbers now, and we're going to take random numbers and they could be three digit, four digit, five digit numbers, and we're going to go through and we're going to get those random numbers. And we're going to assign them one by one to every one of the 370 elements in the list. That is, we're going to assign a different random number to every element. And then what we're going to do is take that list now, that now has not only the sequence number and the ID number, but now has this random number assigned to every case. And we're going to sort the list by the random number. We're going to take it and go from the smallest random number assigned to the large random number assigned. And then our sample operation is to take the first lowercase n elements in that sorted list. Because they're randomly ordered, I could take the first 20. I could take the second set of 20. I could take the last 20. I could take every tenth one until I get to my 20. It doesn't matter, because they're in random order. This process, if repeated again and again, will yield a simple random sample in the same way. It's without replacement. No element's going to get selected more than once because we're just taking the 20 elements from the beginning of the list. It also turns out to be equal probability selection method. That's E-P-S-E-M, epsem. Now it would be a little hard to do this for 370 cases, to go through and grab random numbers off that table. So instead, what we might do is do it through a process of using the numbers, generating numbers through a machine process. Some kind of a black box mechanism that does that, some kind of a key that's generating numbers. I don't know it you ever had one of these for secure login, where it's always showing a new number there. And when it comes time to enter in, in order to log in, a certain six digit sequence, you gotta look at that key and then enter those six digits in exactly as they appear. And then you enter in. Randomly generated in that way. Well, we can do that through our selection mechanisms, and I show a formula here, a piece of machine code. And I'm going to expand this just to say that we could generate those random numbers for each case by machine much more quickly than we can do it by hand. If we had a way to tell the software system, here's how you generate these things. And so, here's a way to generate these random numbers. We could use a function inside each of our pieces of software, our statistical software, called a random number generator. There are a number of different ones that we've mentioned before. We would use the uniform random number generator, and we would tell it, look, we want uniform random numbers. We want random numbers that are between 0 and 1, we want a sequence of them. And we want you to give us the next number in the sequence when you see this for this case. And we want to make sure that you start at a certain beach in a column in a row and be put in a seed, so you see the URAN and then (0718. The 0718 is the seed, it's the starting point. Start of page 7 and column 18, and just start generating the numbers from that point. And then as you're generating those random numbers, between 0 and 1, multiply it by 500. Now what that's going to do is take the numbers between 0 and 1 and expand them to a set of numbers from 0 up to 500. But they're going to have decimals in them, because those random numbers from 0 to 1 have got a lot of decimals in them anyway. And so [COUGH] It's going to expand the numbers from 0.00000 to 499.9999. And we don't need the decimals. So we're going to apply to a function, truncate the decimals, chop them off. And then we've now got a random number that is between 0 and 499. Not 500, because 500 wouldn't actually get generated in this process. But 0 to 499, we don't want 0 to 499, we want 1 to 500, so we're going to add 1 to it, and that's our random number. That's a random number generating mechanism. But it's built around the idea of the uniform random number generator. That's the key, that URAN function is what we're going to use to assign a random number to every case and it'll be just a random number from 0 to 1, I don't care whether it's from 1 to 500. We just care that it's from 0 to 1 and that's what we use in the sorting. Okay, so that's another way to do simple random sampling. There's another way. Now these are all different processes, procedures, but they yield the same result. They yield a set of samples that we would label simple random. Here we're going to do a sample of size lower case n from capital N again, 20 from our 370. We're going to use random numbers from a table. And we're going to match them to the unique numbers assigned to each case. And we're going to review the process until we have m frame elements selected, 20 frame elements selected. We're going to check for duplicates in the sample. So we've got a sample of 20, we've got them all generated, and we look to see, and lo and behold, of the 20 cases, 2 of them are the same. Well, reject the whole list. Reject that sample and draw another of size 20. We keep doing that until we get a set of 20 that are unique, distinct. This is a very expensive process. We wouldn't do this by hand. We'd let a machine do it. We're going to generate samples that allow duplication. But when we get duplication, we reject the sample. We only keep the sample if it doesn't have duplicates in it. This gives us, in the end, a without replacement sample that is epsem, and is simple random. As a matter of fact, what we're doing is using a broader class of samples, the unrestricted random samples, the ones that include the duplicates in it. And then filtering out the ones that have duplicates, filtering those out of the system and only retaining the ones that don't. We're comparing a restricted or simple random sample versus an unrestricted. And hence a symbol in the lower left, actually this is the symbol sometimes you see on the highway with the multiple stripes through it, which says all restrictions removed. Unrestricted random samples, and we find those that have the restrictions we're interested in. This is not an uncommon sampling technique, but again, what happens that we do is we've seen three different ways to draw these samples. The direct approach we did first, where we deliberately searched for duplicates. The approach where we sorted the list. This approach where we sort the samples, we select out the samples that have the right properties. There are lots of possible techniques. But all of them lead to the same set of possible samples that we could have. And it turns out that any procedure that yields a fixed sample size n and for which every element of the population has the same probability of selection. And every combination of size lower case n has the same probability selection, that's a simple random sample. That's the formal definition. That's impossible to work with, but we just have three different ways to implement it. It turns out that process has a huge number of possible samples. And this is going to turn out to be very important for us. This definition is unmanageable practically, but it opens the ground to explore the possibility, the range of possible samples we can have. There's a huge number of them. Here you see, under our second bullet, all sets of size n distinct elements from capital N, pick one. That's what we're doing. How many possible distinct samples are there? Well, there's a formulation for doing this, something called combinatorics, and some of you may be familiar with this from your work in other areas. In which, we want to choose lowercase n from capital N. We want to choose 20 from 370. And that can be expanded, that kind of an expression, you can do that counting, by taking the kind of expression shown there. You'll see a fraction, a ratio, with a 370, exclamation mark. The 370 exclamation mark means 370 times 369 times 368 times 367 times 366 all the way down to 1, divided by (370-20)!, that is 350. 350 times 349 times 348, and so on, by the way which cancels the stuff in the numerator, right? And also divided by 20 factorial. Well, if you do this number, it's huge. You can see what I've got there. You've got a huge number of possible samples that can be selected. One large set of outcomes from this process. And what we're doing is choosing one of them by these techniques. We have three different techniques and we've chosen one sample from any one of those techniques to be our simple random sample. This idea of there being many possible samples is going to be important, but it's a large set of outcomes out there, it's virtually infinite, when you start looking at numbers like this. It's like, the currency in the lower half, whenever I think about this, I think about these situations in world history in which there was hyper inflation and people began manufacturing notes that had lots and lots of zeroes. This was the dinar in Yugoslavia. And this note had to be manufactured because the prices had escalated for various kinds of reasons, so quickly and so dramatically, that we needed really big numbers to keep track of things to even buy a loaf of bread. Well, here we have really big numbers because this process has a vast number of possible outcomes. Okay, so what practical use is it? Well, if we can use those practical techniques we talked about in the beginning for simple random sampling, it can be a widely used approach, process for simple problems, but it's rarely used by practitioners now. People who do this kind of thing all the time in isolation. To just do a simple random sample is a little bit complicated for lay administration. I wouldn't recommend it to a lot of people who've not done it before. It's too easy to make mistakes, and then you worry about whether you've got the right mistake and whether the mistake made any difference. Oftentimes, it doesn't. But there are more efficient methods of selection available, easier to apply methods. Its appeal is that it relies only on randomization. And so people who don't usually do a lot of sampling might do simple random sampling as a first resort. People who do a lot of it don't do this as a first resort. They use other techniques that give them better properties in their results. So simple random sampling, for the practitioners, is a tool that's to be used in conjunction with other methods. A random sample of elements within a group. That idea of stratification that we talked about, that we're going to talk about more in upcoming units. Or a random sample of the groups where we're doing simple random sampling. What we're attempting to do is do things in a form with simple random sampling that we don't ordinarily do. My illustration at the lower left is, somebody came up with decimal time. Instead of our 24-hour clock or our 12-hour clock twice a day, dividing the day into 10 hours, and just having that clock there. No, no, that seems straightforward, but nobody does that. And that's what is true here for simple random sampling. It's a nice idea, but there are better ways. Better ways because of the need to subdivide our time into intervals. 12 is a better number than 10 for that purpose. So, it's just that simple random samples are not quite the best way to do it, there's certainly an acceptable way to do it. But I'm going to encourage us as we move along into other techniques to think about other ways to draw samples, that will be better than simple random sampling. Okay, so that's it for simple random sampling. Simple random sampling gives us our bottom line here, is that it gives us a large set of possible samples. There are a large number of them out there. And what we have to start thinking about is, well gee, we only got one of them. So does it say anything about our quality? And that's something that we're going to look at, is that distribution of our estimates from our sample, across all of those possible samples. We're going to have to imagine this. We're going to imagine what that distribution looks like, and then see how good our sample is, on the basis of what that distribution looks like. It's going to take some time to think that thing through. But we'll do that in the upcoming lecture, where we're going to look at those kinds of questions next. Thank you.
