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

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

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.

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

How about family of three? Num. Girlschild #1child #2child #3 0BBB 1BBG 1BGB 1GBB 2BGG 2GBG 2GGB 3GGG

Probability distribution of number of girls

How about a family of 10?

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

Normal distribution Same shape, if you adjusted the scales CA B

Coin toss Toss a coin 30 times Tabulate results

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

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.

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.

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.

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

But the sampling distribution of mean income over many samples is normal Sampling Distribution of Income, 1992 (thousands) Number of samplesNumber of samples Number of samples

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

s == = = Example of Standard Deviation

Standard Deviation and Normal Distribution

Sample Means S.D. = 2.02 Mean of means = 41.0 Number of Means = 21 Distribution of Sample Means with 21 Samples Frequency

Sample Means Distribution of Sample Means with 96 Samples S.D. = 1.80 Mean of Means = Number of Means = 96

Distribution of Sample Means with 170 Samples Frequency Sample Means S.D. = 1.71 Mean of Means= Number of Means= 170

If all possible random samples of size N are drawn from a population with mean x and a standard deviation s, then as N becomes larger, the sampling distribution of sample means becomes approximately normal, with mean x and standard deviation. The Central Limit Theorem

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.

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

Population inferences can be made...

...by selecting a representative sample from the population

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.

Systematic Random Sampling

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)

Disproportionate Stratified Sample

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