ProbabilitySampling

Probability Sampling

• Simple Random Sample (SRS)• Stratified Random Sampling• Cluster Sampling

• The only way to ensure a representative sample is to obtain it through probability sampling.

What you Need

• In order to obtain ANY kind or Probability Sample you need a SAMPLING FRAME

• A Sampling Frame is a list of all the sampling units.

Sampling Frame

• Ideally the sampling frame contains all elements in the population.

Population

CHHS students

Cars in the city of Salinas

Monterey County Residents

Sampling Frame

A list of all current CHHS students*

A list of all cars owned by residents of Salinas*

A list containing all Monterey County Residents*

*Al list containing groupings of these elements may also work as we will see in cluster sampling

Simple Random Sample

Definition:If a sample of size n is drawn from a population in such a way that every possible sample of size n has the same probability of being selected then this sampling procedure is called Simple Random Sample.

• This is the simplest (in theory).• All of the statistical tools you learned in your basic stats course

assume a SRS.

Sampling frame of CHHS students

Simple Random Sample

Lower division

Higher division

To obtain a Simple Random Sample just number the observations in your sampling frame and use a random number generator to obtain as many observations as you need.

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Sampling frame of CHHS students

Simple Random Sample

Here I obtained a sample of 10 students. I asked Excel to give me 10 random numbers between 1-100*. Excel gave me:

3;16;23;28;44;77;81;90;94

* I used the excel function =RANDBETWEEN(1,100) 10 times and it gave me 10 random numbers between 1 and 100. If a number came twice I asked for a different number

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This is a representative sample!

Stratified Random SampleDefinition:A Stratified Random Sample is one obtained by separating the population elements into non-overlapping groups, called strata, and then selecting a simple random sample from each stratum.

• It sometimes will reduce the cost of collecting data as you can divide population elements into convenient groupings

• It is better than Simple Random Sampling when we are interested in estimating population parameters by subgroups (strata).

• In general a Stratified Random Sample will give you lower estimation error (specially when the strata are homogeneous)

Sampling frame of CHHS students

Stratified Random Sample

Stratum1 Lower divisionN1=30

Stratum 2Higher divisionN2=70

To obtain a Stratified Random Sample you need a sampling frame that identifies the population into non overlapping strata. Then you collect simple random samples from each strata.

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How many observations from each stratum?

• Many researchers prefer proportional allocation.

• Under proportional allocation the number observations selected within each stratum should be proportional to the size of the stratum in relation to the population.– e.g. if a stratum represents 30% of the population then observations

collected from that stratum should represent 30% of the sample.

Sampling frame of CHHS students

Stratified Random Sample

Stratum 1 Lower divisionN1=30

Stratum 2Higher divisionN2=70

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Here I need a sample of 10 students.

Stratum 1 has 30% of the population so sampled observations from stratum 1 should represent 30% of the sample (3 observations)

Stratum 2 has 70% of the population so sampled observations from stratum 2 should represent 70% of the sample (7 observations)

Sampling frame of CHHS students

Stratified Random Sample

Stratum1 Lower divisionN1=30

Stratum 2Higher divisionN2=70

Obtaining a random sample of 3 observations from stratum 1 I got observations3;16;23

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Obtaining a random sample of 7 observations from stratum 2 I got observations34;37;52;77;81;90;94

Cluster SamplingDefinition:A cluster sample is a probability sample in which each sampling unit is a collection or cluster of elements.

• It usually is cheaper than Simple or Stratified Random sampling

• It is extremely useful when the sampling frame has only clusters of elements.• e.g if you wanted a list for all CSUMB students as a sampling frame you

will not get it. However, you can easily obtain a list of all classes (where students are grouped or clustered) offered in a given semester.

• The census uses cluster sampling. They don’t have a list of all individuals but they do have a list of all city blocks

Cluster Sampling

• In cluster sampling we usually use SRS or Stratified Random Sampling to obtain a sample of clusters and then survey all the elements within a cluster.

Sampling frame of CHHS classes

Cluster Sample

First we must randomly select some clusters

And then survey all the students in the selected clusters

Choosing the number of clusters to sample is tricky, there are no rules for it really it depends on how much resources are available and the precision required

CHHS 302

CHHS 385CHHS 300

CHHS496

CHHS405

CHHS 211

CHHS 320 CHHS 400

Total number of clusters =8

CHHS 211

CHHS 302

Sampling frame of CHHS classes

Cluster Sample

In this example I randomly picked 3 clusters out of the 8:CHHS 385CHHS 302CHHS 211

Then I need to survey all the students in each of the 3 clusters. That gives me a total of 41 students.

CHHS 300

CHHS496

CHHS405

CHHS 320 CHHS 400

Total number of clusters =8

Summary• The topic of probability sampling is vast.

• Probability sampling is the ONLY way to obtain representative samples.

• We discussed three basic types of probability sampling each of them has its pros and cons.

• It should be clear that in order to do any of the methods we discussed here you need a “sampling frame.”

• All the statistical techniques you will learn in this course assume a simple random sample.

• Other techniques are needed to make inferences about a population using stratified or cluster sampling.

Next lectures

• Check the video on how to draw probability samples from a sampling frame in Excel.

• What happens when you do not have a sampling frame?

• We turn to non-probability sampling next.