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Simple Random and Systematic Random Sampling

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This tutorial talks about simple random samples. Simple random samples are one type of random sampling. With a Simple Random Sample, it's occasionally abbreviated SRS. In it every possible sample of size n, so whatever size you set your sample to be-- 2, 5, 6, 100-- has an equal probability of being selected.

So for example, if you put the name of everyone in the office in a hat, then shake the hat, and draw out two names. The population here is our office and everybody in it. And the sample is the two names that are drawn. So with a simple random sample, every possible sample of size n has an equal probability of being selected. So every possible sample of two names has an equal probability of being selected. So in this Simple Random Sample, SRS, that is true. Every possible set of two names has an equal probability of being selected.

Here's another example. In this case, our population is the 12 months. Rather than actually cut out the 12 months and put them in a hat and shake it up, instead we could assign numbers to them, and then draw random numbers to represent each month. Now with months, it makes sense to have January be 1 and February be 2, so we don't need to actually assign numbers. But if you're doing something with people, you could go through an assign the numbers to people and then draw some random numbers and use those to select people you want.

So in this example here, I used my random number generator, which I'll talk in a minute, and pulled the numbers 3, 6, and 11. So in that case, the 3 corresponds to March. So that one is in my simple random sample. The 6-- so 3, 4, 5, 6-- responds to June-- corresponds to June. And the 11 corresponds to November. So that is my simple random sample that I've drawn.

Any possible group of three is possible. So I could have had October, November, December, if that's what the random numbers that came up with. So every possible sample of size 3 has an equal probability of being drawn.

Now what if our population looked like this? If we had all of these women and just three men, could it be possible to have the three men be drawn as our simple random sample? And the answer to that is yes. It is possible to draw a simple random sample of those three men. If we're drawing a sample of three, every possible combination of three people is possible. This is one of those possible combinations. So it is possible to draw this one.

However, it's not very likely to draw this set of three men, because of all of the women in our population. It's more likely that we would draw a sample set that involves some women in it. But this is definitely possible and could be simple random sample for this population.

Now before I talked about generating a random number. There a couple of ways of going about doing this. One option was a random number table. Here's an example up here of what a random number table or a selection from it could look like. It's a table with the digits 0 through 9 arranged randomly. Each digit has no predictable relationship to the number that is before or after it and each digit appears with roughly equal appearance.

So knowing that there's a 7 right here you don't really know anything about what's going to come next. And you know that throughout the entirety of the table, all the digits are pretty evenly distributed.

Now there's another possibility called a random number generator. And that's a couple of different things. Typically, I think of a computer that creates a list of random numbers. There is one that uses atmospheric noise to help generate these random numbers. Others use programs and different things like that. Also dice and spinners are ways of generating random numbers, if you don't have a computer or a random number table.

So this has been your tutorial on simple random samples.

Source: Dictionary Table created from data found @ http://juliantrubin.com/encyclopedia/mathematics/dictionary_statistics.html

This tutorial covers systematic random sampling. A systematic random sample is one of many kinds of random sampling. In this type, you're taking every blankth person to be part of the sample. In the blank, you fill in whatever number. So you could take every 4th person, every 10th person, every 100th person, and then that person is included in your sample. And you count up to the next hundredth person, and that person also goes into your sample.

The number you choose doesn't really matter. It's going to depend on how large your population is and how many people you need to survey. Now an advantage of systematic random sampling is that it's sometimes easier to do than a real simple random sample. You don't have to assign everyone a number. In this case, you're just counting off people as they pass by you or you're counting off people in a group. A disadvantage is that it's sometimes not actually random. If the people you're looking at are in an order, then it's no longer random.

Let's look at a simple example. In this example, we're taking every fourth person that walks by us on the street to take our survey. So we count across the people on the street. 1, 2, 3 people pass us, then the fourth person goes in the survey. Then another 1, 2, 3 people pass by and that next person goes in the survey. We let another 1, 2, 3 people go by, and the next person is in the survey.

This is easy enough to do because we can take a sample from any population anywhere without having to prepare by assigning people numbers. Let's look at an example.

In this example here, we're looking at when a store prints out a receipt and at the bottom of the receipt, it has a survey. Usually there's some sort of incentive for filling out this survey. Now there's no particular order to the customers that are exiting the store. So this systematic sample ends up being random. Because there's no predictable pattern to how the customers are exiting, there's no pattern to who that 10th person is that gets that survey on the bottom of the receipt. So this example ends up being a random sample.

Let's look at an example that ends up being not random because the people are in an order. When people are in an order, the sample ends up being not random. I like to think of when I was a kid and we were playing dodge ball in gym class. The gym teacher would line everybody up in a circle and start to count people off.

Now let's look at a little more specific. Let's imagine that this person here is my best friend. Now obviously, I would stand next to my best friend because we want to be as close together at all times. So when the gym teacher starts to count off by threes to make the teams, I realized pretty quickly that my best friend and I weren't going to be on the same team. So I realized I could move around the circle to be in a new spot to end up on the same team.

So if we start with one, 2, 3. And then we go again, 1, 2, 3. 1, 2, 3. In order to be on the same team as my friend, I would have to be a number 1. So I would move around to this spot here. Well, in this case, we're picking teams for gym class. We're not picking people for a survey. It's similar enough to look at it.

So when people can either determine their own order and decide whether or not they're going to be in the survey because of how they're getting ordered or if people are arranged by height or intelligence or even last name, it's no longer random, if the order effects what sample gets drawn. So this is no longer random and potentially no longer representative.

Let's look at one more example. This is a real example of someone who wanted to use a systematic random sample in order to evaluate how many words have been in dictionaries over time. First, they started with a random page. Maybe they used a random number table. Let's say they came up with 7.7. So they're going to start with page 7. And then they're going to take every fiftieth page to be part of their sample. So you'd take page 7 and then 57, 107, 157, and so forth, until they reached the end of the dictionary.

On those selected pages, they would count how many head words were on the page. Once you find out how many words are on each page that you've selected the, 7th 57th, 107th, 157th, you can be pretty sure that those were a good representation of what's in the dictionary. Then you can average those numbers all together to find out the average number of words per page.

Once you know how many words are on each page, if you multiply them by how many pages there are, you'd get the total, the total number of words in that dictionary. Let's look at the results of what the person found.

In the first dictionary they looked at, one from 1941, there were 16 words on each page. Then the dictionary had 1,283 pages for a total of 20,849 words in the dictionary. Now they repeated this for eight more dictionaries, ending with one printed in 2010. In 2010 there are 25 words per page 1,796 pages for a total of 45,956 words in the dictionary.

The rest of the dictionaries in the sample show the same pattern of increasing in size over time. So here's a way where a systematic random sample can save us a lot of time. Instead of counting every single one of those 45,956 words, we can take a sample of the pages. We can count how many words are on those pages, find an average, and multiply in order to draw a conclusion.

This has been your tutorial on systematic random sampling. There are several ways in which you can take a systematic random sample and still have the sample be random. If however, your population is ordered in any way, the sample will no longer be random and potentially no longer be representative.