1. SAMPLING
Purposes
Representativeness
“Sampling error”

2. Review: essential definitions
Population (N=size)
Largest group to which we intend to project (apply) the findings of a study
All prisoners in Jay’s prison / all students in his class
“Parameter” - any statistic (e.g., mean) of a population
Sample (n=size)
Any subgroup of the population
Samples intended to represent a population must be selected in special ways (will come up later)
Unit of analysis
The “container” for the variables
Here, the variables under study are sentence length and type of crime (property or violent)
What “contains” them? Prisoners!
Case
A single occurrence of a unit of analysis
Here, it’s any one prisoner
Cases are “members” or “elements” of the population from which one or more samples are drawn
Sampling frame
A list of all “elements” or members of the population
Jay’s prison / Jay’s class
Population
Sample

3. Purposes & representativeness
Purposes of sampling
Descriptive: Describe characteristics of a population without having to measure every member (e.g., age, height, gender)
Explanatory: Help test hypotheses of cause and effect (e.g., gender determines height)
Representativeness: Samples should accurately reflect, or represent, the population from which they are drawn
We will be exploring ways to make samples “representative”
If a sample is representative, we can apply findings from that sample - “make inferences” - to the population from which the sample was drawn 21 28 70 M 68 62 F 26 63 23 22 67 24 73 29 25 65
Population parameters
Sample statistics 10
24.00

4. Sampling error
Sampling error: Unintended differences between a population parameter and the equivalent statistic from an unbiased sample
Inevitable result of sampling
Try it out in class! Calculate the population mean for age.
Then take samples of different size and calculate their mean.
Any difference between a population parameter and a sample statistic is “sampling error.” It should decrease as sample size increases
Rule of thumb
To minimize sampling error, sample size should be at least 30 for populations up to about 500
For larger populations sample sizes should be larger
Population parameters
Sample 1 statistics 3
Sample 2 statistics 10

5. PROBABILITY SAMPLING With/without replacement Simple random sampling
Stratified random sampling / proportionate
Stratified random sampling / disproportionate

6. Sampling with / without replacement
With replacement: Return each case to the population before drawing the next
Makes it possible to redraw the same case (not good)
Keeps the probability of a case being drawn the same from beginning to end (good)
Without replacement: Drawn cases are not returned to the population
Probability of undrawn cases being selected increases as cases are drawn
Sampling without replacement is by far the most common
Most sampling frames are sufficiently large so that as cases are drawn changes in the probability that any particular case will be drawn are small X X

7. Simple random sampling
In simple random sampling we draw at random from the entire population 21 28 70 M 68 62 F 26 63 23 22 67 24 73 29 25 65

8. Using simple random sampling to describe a population
Population: 200 inmates Mean sentence: 2.94 years
Assignment
Draw a random sample of 10 and compare its mean to the population parameter. Then do the same with a random sample of 30.
How much error is there? Does it change with sample size?
Data from
Jay’s correctional center
Koko Wachtel, warden
Frequency (# prisoners)
Sentence length in years

9. Stratified random sampling - proportionate
(M=21 F=10)
n=20
(M=14 F=6)
n=14
n=6 F M
If you did a good job randomly sampling, the size of each group - its “n”, or number of cases - should be roughly proportionate to that score or value’s proportion in the population
Proceed with your analysis
“Stratify” - group these cases according to their value or score on your variable of interest
These groups are called “strata” (sing., “stratum”)
Draw a random sample (say, n=20) from the population

10. Stratified random sampling - disproportionate
First, designate a variable of interest (gender) Separate the population into subgroups (“strata”) that correspond with the variable’s values (M, F)
Draw random samples of equal size from each subgroup
Proceed with your analysis

11. Using stratified random sampling to describe a population
Population 200 inmates; mean sentence 2.94 years
Assignment
Draw a random sample of 30 from each stratum and compare its mean to the corresponding population parameters.
How much error is there?
Property crimes: Mean sentence: 2.88
Violent crimes: 50 Mean sentence: 3.12

12. Using stratified random sampling to test a hypothesis - exercises1 2
Hypothesis: A pre-existing personal relationship between criminal and victim is more likely in violent crimes than in crimes against property You have full access to crime data for Sin City. These statistics show that in 2014 there were 200 crimes, of which 75 percent were property crimes and 25 percent were violent crimes. For each crime, you know whether the victim and the suspect were acquainted (yes/no).
Hypothesis: Male cops are more cynical than female cops
Sin City Police Department has 200 officers; 150 are male and 50 are female.
1. Identify the population. 
How would you sample proportionally?
How would you sample disproportionately?
In this example which of the above is preferable? Why?

13. randomly select 30 cases (15% of the population)
Stratified proportionate random sampling
Hypothesis: A pre-existing personal relationship between criminal and victim is more likely in violent crimes than in crimes against property
Sin City 200 crimes in 2014
50 violent (25 %)
150 property (75 %)
randomly select 30 cases (15% of the population)
(expect 7.5 violent – 25%)
(expect 22.5 property – 75%)
Compare proportions within each where suspect and victim were acquainted
BUT: The frequency (number of cases) for violent crime is very small!

14. randomly select 30 cases from each category
Stratified disproportionate random sampling
Hypothesis: A pre-existing personal relationship between criminal and victim is more likely in violent crimes than in crimes against property
Sin City 200 crimes in 2014
50 violent (25 %)
150 property (75 %)
randomly select 30 cases from each category
30 property
30 violent
Compare proportions within each where suspect and victim were acquainted
Note: don’t recombine these into a single sample to compute a mean!

15. Hypothesis: Male cops are more cynical than female cops
Stratified proportionate random sampling
Hypothesis: Male cops are more cynical than female cops
150 male (75 %)
Sin City 200 officers
50 female (25 %)
randomly select 30 officers
expect 22.5 males
expect 7.5 females
Compare average cynicism scores
BUT: The frequency (number of cases) for females is very small!

16. Hypothesis: Male cops are more cynical than female cops
Stratified disproportionate random sampling
Hypothesis: Male cops are more cynical than female cops
Sin City 200 officers
150 male (75 %)
50 female (25 %)
randomly select 30 officers from each stratum
30 males
30 females
Compare average cynicism scores
Note: don’t recombine these into a single sample to compute a mean!

17. OTHER SAMPLING TECHNIQUES
Quasi-probability sampling:
systematic sampling, cluster sampling
Non-probability sampling

18. Quasi-probability sampling
Systematic sampling
Randomly select first element, then choose every 5th, 10th, etc. depending on the size of the sampling frame (number of cases or elements in the population)
If done with care can give results equivalent to fully random sampling
Caution: if elements in the sampling frame are ordered in a particular way a non-representative sample might be drawn
Cluster sampling
Method
Divide population into equal-sized groups (clusters) chosen on the basis of a neutral characteristic
Draw a random sample of clusters. The study sample contains every element of the chosen clusters.
Often done to study public opinion (city divided into blocks)
Rule of equally-sized clusters usually violated
The “neutral” characteristic may not be so and affect outcomes!
Since not everyone in the population has an equal chance of being selected, there may be considerable sampling error

19. Non-probability sampling
Accidental sample
Subjects who happen to be encountered by researchers
Example – observer ride-alongs in police cars
Quota sample
Elements are included in proportion to their known representation in the population
Purposive/“convenience” sample
Researcher uses best judgment to select elements that typify the population
Example: Interview all burglars arrested during the past month
Issues
Can findings be “generalized” or projected to a larger population?
Are findings valid only for the cases actually included in the samples?

20. Practical exercise: SYSTEMATIC SAMPLING

21. Class assignment - systematic sampling
Hypothesis: Higher income persons drive more expensive cars - Income  Car Value
Independent variable: income
Categorical, nominal: student or faculty/staff
Dependent variable: car value
Categorical, ordinal: 1 (cheapest), 2, 3, 4 or 5 (most expensive)
Panel assignment (worth 5 points)
Select a panel coordinator
Visit a student lot
Select ten vehicles in each lot using systematic sampling
Use the operationalized car values to code each car’s value
Give each team member a filled-in copy and turn one in per team next week
The copy you turn in must have the printed name and signature of each panelist who participated in collecting data
PLEASE BRING THIS FORM TO EVERY CLASS SESSION!