Probability Sampling

1. Probability Sampling Definitions Simple Random Sampling Stratified Sampling Systematic Sampling

2. Sampling • Sampling is the process of selecting units (e.g., people, organizations) from a population of interest so that by studying the sample we may fairly generalize our results back to the population from which they were chosen

3. The logic of sampling • If all members of a population were identical in all respects (demographic characteristics, attitudes, experiences, etc)- there would be no need for careful sampling • In such a case, any sample would be sufficient • In reality, human being are quite heterogeneous

4. The logic of sampling • To provide useful descriptions of the total population, a sample of individuals from a population must contain essentially the same variations that exist in the population • This is not as simple as it might seem

5. Definitions • Population • Sampling Frame • Sample • Subsample

6. Population • The group you wish to generalize to is often called the population in your study • This is the group you would like to sample from because this is the group you are interested in generalizing to

7. Example of population • Let's imagine that you wish to generalize to urban homeless males between the ages of 30 and 50 in the United States. If that is the population of interest, you are likely to have a very hard time developing a reasonable sampling plan

8. Hidden Population • You are probably not going to find an accurate listing of this population • So we probably should make a distinction between the population you would like to generalize to, and the population that will be accessible to you • We'll call the former the theoretical population and the latter the accessible population • In this example, the accessible population might be homeless males between the ages of 30 and 50 in six selected urban areas across the U.S.

9. Sampling Frame • Once you've identified the theoretical and accessible populations, you have to get a list of the members of the accessible population. • The listing of the accessible population from which you'll draw your sample is called the sampling frame.

10. Phone book • If you were doing a phone survey and selecting names from the telephone book, the book would be your sampling frame • That wouldn't be a great way to sample because significant subportions of the population either don't have a phone or have moved in or out of the area since the last book was printed.

11. Sample • Finally, you actually draw your sample (using one of the many sampling procedures). • The sample is the group of people who you select to be in your study • Notice that I didn't say that the sample was the group of people who are actually in your study.

12. Subsample • You may not be able to contact or recruit all of the people you actually sample, or some could drop out over the course of the study • The group that actually completes your study is a subsample of the sample -- it doesn't include nonrespondents or dropouts.

14. Probability sampling • For a sample to be minimally scientific it should be random, i.e. the elements for the sample should be selected on the random basis out of the will of the sampler or the respondent. • (1) the elements in the sample should be randomly selected, and • (2), the likelihood of each of the elements to be selected should be known.

15. A Population of 100 Folks Babbie 2005

16. A Sample of Convenience: Easy, but Not Representative Babbie 2005

17. Random selection • picking a name out of a hat • tossing coins • rolling dice • choosing the short straw • These days, we tend to use computers as the mechanism for generating random numbers as the basis for random selection

18. Advantages vs disadvantages • Advantages: ideal for statistical purposes • Disadvantages: • hard to achieve in practice • requires an accurate list of the whole population • expensive to conduct as those sampled may be scattered over a wide area

19. Types of Sampling Designs • Simple random sampling (SRS) • Stratified sampling • Systematic sampling • Cluster sampling

20. Simple Random Sampling • A simple random sample gives each member of the population an equal chance of being chosen.  • One way of achieving a simple random sample is to number each element in the sampling frame and then use random numbers to select the required sample  • Random numbers can be obtained using your calculator, a spreadsheet, printed tables of random numbers

21. Random Number Tables • Random number tables consist of a randomly generated series of digits (0-9) • To make them easy to read there is typically a space between every 4th digit and between every 10th row • When reading from random number tables you can begin anywhere (choose a number at random) but having once started you should continue to read across the line or down a column and NOT jump about.

22. Random Number Tables 3963 6234 4088 6556 1637 9197 1339 1536 9459 1798 1459 3505 4046 2747 4452 6733 9336 5452 2235 9320 3073 7157 3722 7971 2577 6517 0776 8292 3113 3019 8912 9125 2409 2575 0309 3941 7314 0608 1563 8033 3680 2231 8846 5418 0498 5245 7071 2597 8046 2438 4283 9511 4351 4208 1514 3473 6807 1829 6948 8046

23. Example • We want to sample two houses from a street containing houses numbered 1 to 48

24. Generate a list of houses (population elements) • 1Kingsley St. • 2 Kingsley St. • 3 Kingsley St. • 4 Kingsley St.. • ………………… • 47 Kingsley St.. • 48 Kingsley St.

25. Random Number Tables 3963 6234 4088 6556 1637 9197 1339 1536 9459 1798 1459 3505 4046 2747 4452 6733 9336 5452 2235 9320 3073 7157 3722 7971 2577 6517 0776 8292 3113 3019 8912 9125 2409 2575 0309 3941 7314 0608 1563 8033 3680 2231 8846 5418 0498 5245 7071 2597 8046 2438 4283 9511 4351 4208 1514 3473 6807 1829 6948 8046

26. Example • Here is an extract from a table of random sampling numbers: • 3680 2231 8846 5418 0498 5245 7071 2597 • If we wanted to sample two houses from a street containing houses numbered 1 to 48 we would read off the digits in pairs36 80 22 31 88 46 54 18 04 98 52 45 70 71 25 97 and take the first two pairs that were less than 48, which gives house numbers 36 and 22.

27. Example • If we wanted to sample two houses from a much longer road with 140 houses in it we would need to read the digits off in groups of three:368 022 318 846 541 804 985 245 707 125 975and the numbers underlined would be the ones to visit: 22 and 125.

28. A Population of 100 Folks Babbie 2005

29. A Simple Random Sample Babbie 2005

30. Stratified Random Sampling • Stratified Random Sampling, also sometimes called proportional or quota random sampling, involves dividing your population into homogeneous subgroups and then taking a simple random sample in each subgroup. • The strata should be mutually exclusive: every element in the population must be assigned to only one stratum. • Objective: Divide the population into non-overlapping groups (i.e., strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3 + ... + Ni = N. Then do a simple random sample of f = n/N in each strata.

31. Stratified Random Sampling • If the population consists of 60% in the male stratum and 40% in the female stratum, then the relative size of the two samples (three males, two females) should reflect this proportion.

32. Doing Stratified Sampling • Divide the total population into homogenous groups based on some variable • The researcher assumes that any 2 items in the strata are more similar, i.e. the stratum is homogeneous, than any 2 items from different strata. • Stratification increases the efficiency (i.e. smaller sampling error) by reducing the variation of elements between the stratum.

33. Example • We want to sample two houses from east side and two houses from west side of a street containing houses numbered 1 to 48

34. 1 Kingsley St. 3 Kingsley St. 5 Kingsley St. 7 Kingsley St.. ………………… 45 Kingsley St.. 47 Kingsley St. 2 Kingsley St. 4 Kingsley St. 6 Kingsley St. 8 Kingsley St.. ………………… 46 Kingsley St.. 48 Kingsley St. Generate a list of houses (population elements) West Side East Side

35. Random Number Tables 3963 6234 4088 6556 1637 9197 1339 1536 9459 1798 1459 3505 4046 2747 4452 6733 9336 5452 2235 9320 3073 7157 3722 7971 2577 6517 0776 8292 3113 3019 8912 9125 2409 2575 0309 3941 7314 0608 1563 8033 3680 2231 8846 5418 0498 5245 7071 2597 8046 2438 4283 9511 4351 4208 1514 3473 6807 1829 6948 8046

37. Systematic Random Sampling • Here are the steps you need to follow in order to achieve a systematic random sample: • Number the units in the population from 1 to N • Decide on the n (sample size) that you want or need • k = N/n = the interval size • Randomly select an integer between 1 to k • Then take every kth unit in your sample frame

38. 1Kingsley St. 2 Kingsley St. 3 Kingsley St. 4 Kingsley St. 5 Kingsley St. 6 Kingsley St. ………………… 15 Kingsley St. ………………… 27 Kingsley St. ………………… 39 Kingsley St. ………………. 48 Kingsley St. N=48 n=4 K=N/n=12 Random start = 3 Generate a list of houses (population elements)

39. Example • Let's assume N=100 people • You want to take a sample of n=20. • To use systematic sampling, the population must be listed in a random order. • The interval size, k, is equal to N/n = 100/20 = 5. • Now, select a random integer from 1 to 5. • In our example, imagine that you chose 4 • Now, to select the sample, start with the 4th unit in the list and take every k-th unit (every 5th, because k=5). You would be sampling units 4, 9, 14, 19, and so on to 100 and you would wind up with 20 units in your sample.

41. A Stratified Systematic Sample with a Random Start • Step 1: Stratification takes place (grouping members of the population into relatively homogeneous subgroups before sampling) • Step 2: Systematic sampling is applied within each stratum. Babbie 2005

42. Example (3f and 6m) • Smith (f) • Sanders (f) • Larson (m) • Barton (m) • Briton (f) • Bell (m) • Dubin (m) • Molin (m) • Moore (f) • Vince (m)

43. Female Smith Sanders Briton Moore Male Larson Barton Bell Dubin Molin Vince Stratification (40% vs 60%) N=5 Female=40% (n=2) Male= 60% (n=3)

44. Systematic sampling from each strata • Female (k=2) • Male (k=2)