Sampling and Sampling Distributions

1. Sampling and Sampling Distributions • Aims of Sampling • Probability Distributions • Sampling Distributions • The Central Limit Theorem • Types of Samples

2. Aims of sampling • Reduces cost of research (e.g. political polls) • Generalize about a larger population (e.g., benefits of sampling city r/t neighborhood) • In some cases (e.g. industrial production) analysis may be destructive, so sampling is needed

3. Probability • Probability: what is the chance that a given event will occur? • Probability is expressed in numbers between 0 and 1. Probability = 0 means the event never happens; probability = 1 means it always happens. • The total probability of all possible event always sums to 1.

4. Probability distributions: Permutations What is the probability distribution of number of girls in families with two children? 2 GG 1 BG 1 GB 0 BB

6. How about family of three?

7. Probability distribution of number of girls

8. How about a family of 10?

9. As family size increases, the binomial distribution looks more and more normal.

10. Normal distribution Same shape, if you adjusted the scales B A C

11. Coin toss • Toss a coin 30 times • Tabulate results

12. Coin toss • Suppose this were 12 randomly selected families, and heads were girls • If you did it enough times distribution would approximate “Normal” distribution • Think of the coin tosses as samples of all possible coin tosses

13. Sampling distribution Sampling distribution of the mean – A theoretical probability distribution of sample means that would be obtained by drawing from the population all possible samples of the same size.

14. Central Limit Theorem • No matter what we are measuring, the distribution of any measure across all possible samples we could take approximates a normal distribution, as long as the number of cases in each sample is about 30 or larger.

15. Central Limit Theorem If we repeatedly drew samples from a population and calculated the mean of a variable or a percentage or, those sample means or percentages would be normally distributed.

16. Most empirical distributions are not normal: U.S. Income distribution 1992

17. But the sampling distribution of mean income over many samples is normal Number of samples Number of samples 18 19 20 21 22 23 24 25 26 Sampling Distribution of Income, 1992 (thousands)

18. Standard Deviation Measures how spread out a distribution is. Square root of the sum of the squared deviations of each case from the mean over the number of cases, or

19. = 129.71 = s = = Example of Standard Deviation 2 2

20. Standard Deviation and Normal Distribution

21. Distribution of Sample Means with 21 Samples 10 8 6 4 2 0 S.D. = 2.02 Mean of means = 41.0 Number of Means = 21 Frequency 37 38 39 40 41 42 43 44 45 46 Sample Means

22. Distribution of Sample Means with 96 Samples 14 12 10 8 6 4 2 0 S.D. = 1.80 Mean of Means = 41.12 Number of Means = 96 Frequency 37 38 39 40 41 42 43 44 45 46 Sample Means

23. Distribution of Sample Means with 170 Samples 30 20 10 0 S.D. = 1.71 Mean of Means= 41.12 Number of Means= 170 Frequency 37 38 39 40 41 42 43 44 45 46 Sample Means

24. The standard deviation of the sampling distribution is called the standard error

25. The Central Limit Theorem Standard error can be estimated from a single sample: Where s is the sample standard deviation (i.e., the sample based estimate of the standard deviation of the population), and n is the size (number of observations) of the sample.

26. Confidence intervals Because we know that the sampling distribution is normal, we know that 95.45% of samples will fall within two standard errors. 95% of samples fall within 1.96 standard errors. 99% of samples fall within 2.58 standard errors.

27. Sampling • Population – A group that includes all the cases (individuals, objects, or groups) in which the researcher is interested. • Sample – A relatively small subset from a population.

28. Random Sampling • Simple Random Sample – A sample designed in such a way as to ensure that (1) every member of the population has an equal chance of being chosen and (2) every combination of N members has an equal chance of being chosen. • This can be done using a computer, calculator, or a table of random numbers

29. Population inferences can be made...

30. ...by selecting a representative sample from the population

31. Random Sampling • Systematic random sampling – A method of sampling in which every Kth member (K is a ration obtained by dividing the population size by the desired sample size) in the total population is chosen for inclusion in the sample after the first member of the sample is selected at random from among the first K members of the population.

32. Systematic Random Sampling

33. Stratified Random Sampling • Proportionate stratified sample – The size of the sample selected from each subgroup is proportional to the size of that subgroup in the entire population. (Self weighting) • Disproportionate stratified sample – The size of the sample selected from each subgroup is disproportional to the size of that subgroup in the population. (needs weights)

34. Disproportionate Stratified Sample

35. Stratified Random Sampling • Stratified random sample – A method of sampling obtained by (1) dividing the population into subgroups based on one or more variables central to our analysis and (2) then drawing a simple random sample from each of the subgroups