**In applications**:

Probability Sampling: Simple Random Sampling, Stratified Random Sampling, Multi-Stage Sampling

- What is each and how is it done?
- How do we decide which to use?
- How do we analyze the results differently depending on the type of sampling?

Non-probability Sampling: Why don't we use non-probability sampling schemes? Two reasons:

- We can't use the mathematics of probability to analyze the results.
- In general, we can't count on a non-probability sampling scheme to produce representative samples.

**In mathematical statistics books** (for courses that assume
you have already taken a probability course):

- Described as assumptions about random variables
- Sampling with replacement versus sampling without replacement

**What are the main types of sampling and how
is each done?**

**Simple Random Sampling**: A simple random sample (SRS) of
size *n* is produced by a scheme which ensures that each
subgroup of the population of size *n* has an equal probability
of being chosen as the sample.

**Stratified Random Sampling**: Divide the population into
"strata". There can be any number of these. Then choose
a simple random sample from each stratum. Combine those into the
overall sample. That is a stratified random sample. (Example:
Church A has 600 women and 400 women as members. One way to get
a stratified random sample of size 30 is to take a SRS of 18 women
from the 600 women and another SRS of 12 men from the 400 men.)

**Multi-Stage Sampling**: Sometimes the population is too large
and scattered for it to be practical to make a list of the entire
population from which to draw a SRS. For instance, when the a
polling organization samples US voters, they do not do a SRS.
Since voter lists are compiled by counties, they might first do
a sample of the counties and then sample within the selected counties.
This illustrates two stages. In some instances, they might use
even more stages. At each stage, they might do a stratified random
sample on sex, race, income level, or any other useful variable
on which they could get information before sampling.

**How does one decide which type of sampling to use?**

The formulas in almost all statistics books assume simple random sampling. Unless you are willing to learn the more complex techniques to analyze the data after it is collected, it is appropriate to use simple random sampling. To learn the appropriate formulas for the more complex sampling schemes, look for a book or course on sampling.

Stratified random sampling gives more precise information than simple random sampling for a given sample size. So, if information on all members of the population is available that divides them into strata that seem relevant, stratified sampling will usually be used.

If the population is large and enough resources are available, usually one will use multi-stage sampling. In such situations, usually stratified sampling will be done at some stages.

**How do we analyze the results differently depending on the
different type of sampling?**

The main difference is in the computation of the estimates of
the variance (or standard deviation). An excellent book for self-study
is *A Sampler on Sampling*, by Williams, Wiley. In this,
you see a rather small population and then a complete derivation
and description of the sampling distribution of the sample mean
for a particular small sample size. I believe that is accessible
for any student who has had an upper-division mathematical statistics
course and for some strong students who have had a freshman introductory
statistics course. A very simple statement of the conclusion is
that the variance of the estimator is smaller if it came from
a stratified random sample than from simple random sample of the
same size. Since small variance means more precise information
from the sample, we see that this is consistent with stratified
random sampling giving better estimators for a given sample size.

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**Non-probability sampling schemes**

These include voluntary response sampling, judgement sampling, convenience sampling, and maybe others.

In the early part of the 20^{th} century, many important
samples were done that weren't based on probability sampling schemes.
They led to some memorable mistakes. Look in an introductory statistics
text at the discussion of sampling for some interesting examples.
The introductory statistics books I usually teach from are *Basic
Practice of Statistics* by David Moore, Freeman, and *Introduction
to the Practice of Statistics* by Moore and McCabe, also from
Freeman. A particularly good book for a discussion of the problems
of non-probability sampling is *Statistics* by Freedman,
Pisani, and Purves. The detail is fascinating. Or, ask a statistics
teacher to lunch and have them tell you the stories they tell
in class. Most of us like to talk about these! Someday when I
have time, maybe I'll write some of them here.

Mathematically, the important thing to recognize is that the discipline of statistics is based on the mathematics of probability. That's about random variables. All of our formulas in statistics are based on probabilities in sampling distributions of estimators. To create a sampling distribution of an estimator for a sample size of 30, we must be able to consider all possible samples of size 30 and base our analysis on how likely each individual result is.

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Mathematical statistics texts almost always says to consider the
X's (or Y's) to be independent with a common distribution. How
does this correspond to some description of how to sample from
a population? **Answer**: simple random sampling with replacement.

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