1. Why sample?
Diversity in populations
Practicality and cost

2. Terms
Population = large group about which conclusions are drawn. Real, but unknown.
Sample = small group that represents population. Real, known.
Sample
Sample
Sample
Sample
Sample
Sample
Population
Sample
Sample

3. Element = individual member of a population.
Sampling unit = element or group of elements selected in a sample.
Unit of analysis = element or group of elements compared in the analysis
The above units can be the same or different.

4. Element, Sampling Unit, Unit of Analysis: Examples
Opinion Survey of UMD Students
Element = individual student
Sampling unit = individual student
Unit of analysis = individual student (student opinions measured)
Survey of family incomes
Element = adult household member
Sampling unit = household or address
Unit of analysis = family (total family income measured)

5. Element, Sampling Unit, Unit of Analysis: More Examples
Voter Polls
Element = individual voter
Sampling unit = telephone number
Unit of analysis = individual voter
(voter opinions measured)
U.S. Census of housing
Element = household or address
Sampling unit = household or address
Unit of analysis = household or address
(# of rooms measured)

6. Sampling frame = list of all the sampling units in the population
Sampling frame = list of all the sampling units in the population. Needed for probability sampling.
Probability sample = researcher knows and controls the probability of selection.
Main advantage: Only probability samples permit accurate estimation of sampling error.

7. Simple Random Sample
Every element in the population has an equal and constant chance of selection
1. Physical sampling with replacement
2. Table of random numbers
3. Random selection by computer
Probability of selection = Sample Size/ Pop. size
Requires list (frame) of all elements in population

8. Systematic Random Sample
Every “kth” element is drawn from a list. (e.g. every 50th name)
1. K = sampling interval = Pop. Size/Sample size (e.g. 5000/100).
2. Random starting point between 1 and K (e.g. 1 and 50).
3. Statistically equivalent to simple random sample)
4. List must be randomly ordered.
5. Convenient, since lists are available for many populations

9. Stratified Random Sample
Population is first divided into groups (strata).
Simple random sample is taken from within each stratum
Separate random samples are combined into a single total sample.

10. Example of stratified sample
Seniors
Sample 2
Juniors
Sample
Sample
UMD Population
Sample 3
Sophomores
Freshmen
Sample 4
Sample 1
Sample 1
Stratified Sample
Sample 2
Sample 2
Sample 3
Sample 3
Sample 4
Sample 4

11. Considerations in Stratified Sampling
Requires knowledge of stratifying variable
Best used when there is much variation between strata in variable being measured (Example: Stratify by year in school if measuring opinions of advising)
Lowest sampling error
Most costly

12. Sampling error = estimated difference between sample value and actual population value (e.g. + 3%)

13. Cluster Sample
Elements in population are naturally grouped together (“clusters”)
Simple random sample of clusters is taken
Every element in selected clusters is studied.
Population:
Sample

14. Considerations in Cluster Sampling
Best when there is little variation between clusters in variable being measured.
Does not require a list of individual elements (only clusters).
May be used to cover large geographic area (smaller areas = clusters)
May be less expensive
Highest sampling error.

15. Multistage Designs Combines two or more sampling designs.
Example: sampling voters in MN
Stage 1: Stratify by geographic area (e.g. county)
Stage 2: Sample census tracts (clusters) in selected counties.
Stage 3: Take SRS of households in each tract.
Commonly used in large, diverse populations
Design is best left to experts!

16. Sampling Why use sampling? Terms and definitions
Probability Sampling Designs
Simple random
Systematic
Stratified
Cluster
Multistage designs
Estimation from samples

17. Estimation from Samples
Find a likely range of values for a population parameter (e.g. average, %)
Parameter = characteristic of a population
Statistic = characteristic of a sample
Statistical inference = drawing conclusions about a population based on sample data
Usually connected with a probability of error.

18. Sampling Distribution
Distribution of results of all possible samples of size N taken from same population
Theoretical, not actually done in practice
Properties of sampling distributions are known to statisticians
Used as basis for inferring from samples to populations

19. Example: estimating proportion of homes with internet access
Suppose population proportion = .62
Take 1 sample of size 200 homes have internet access. Sample p = .60
Can we conclude that the population proportion is .60?
A different sample might produce a different answer

20. What if we took all possible samples?
Most sample proportions would be close to population value
A few would be much higher or lower
Average of sample proportions would be the true population proportion
Distribution would be a bell-shaped curve
% of samples
All possible sample proportions

21. What we know from sampling distribution:
We DON’T know the true population proportion.
We DO know how many sample proportions fall within a given distance of the true proportion.
Sampling error = estimated difference between sample value and actual population value
(example: 95% of sample proportions fall within +
3% of true proportion)

22. How we make an estimate Find sample proportion
Add sampling error (margin of error) on either side
True proportion probably falls within this interval
% of samples
All possible sample proportions
% of samples
All possible sample proportions p p p p

23. Examples of estimates
If 95% of sample proportions (p) fall within + 3% of true proportion, then 95% of all intervals p will contain true population proportion.
If p = .6, we estimate the true proportion is = .57 to .63
If p = .62, we estimate the true proportion is = .59 to .65
If p = .57, we estimate the true proportion is = .54 to .60
If p = If p = .7, we estimate the true proportion is = .67 to .73
95% of the time this procedure yields a correct estimate.