**Chapter 7**

**Sampling**

(Reminder: Don’t forget to utilize the concept maps and study questions as you study this and the other chapters.)

The purpose of Chapter 7 it to help you to learn about sampling in quantitative and qualitative research. In other words, you will learn how participants are selected to be part of empirical research studies.

__Sampling__ refers to drawing a sample (a subset) from a population (the
full set).

- The
usual goal in sampling is to produce a
__representative sample__(i.e., a sample that is similar to the population on all characteristics, except that it includes fewer people because it is a sample rather than the complete population). - Metaphorically, a perfect representative sample would be a "mirror image" of the population from which it was selected (again, except that it would include fewer people).

**Terminology Used
in Sampling**

Here are some important terms used in sampling:

- A
__sample__is a set of elements taken from a larger population. - The
sample is a subset of the
__population__which is the full set of elements or people or whatever you are sampling. - A
__statistic__is a numerical characteristic of a sample, but a__parameter__is a numerical characteristic of population. __Sampling error__refers to the difference between the value of a sample statistic, such as the sample mean, and the true value of the population parameter, such as the population mean. Note: some error is always present in sampling. With random sampling methods, the error is random rather than systematic.- The
__response rate__is the percentage of people in the sample selected for the study who actually participate in the study. - A
__sampling frame__is just a list of all the people that are in the population. Here is an example of a sampling frame (a list of all the names in my population, and they are numbered). Note that the following sampling frame also has information on age and gender included in case you want to draw some samples and do some calculations.

**Random Sampling Techniques**

The two major types of sampling in quantitative research are random sampling and nonrandom sampling.

- The former produces representative samples.
- The
latter does
__not__produce representative samples.

**Simple Random Sampling**

The first type of random sampling is called __simple random
sampling__.

- It's the most basic type of random sampling.
- It
is an equal probability sampling method (which is abbreviated by
__EPSEM__). - Remember that EPSEM means "everyone in the sampling frame has an equal chance of being in the final sample."
- You
should understand that using an EPSEM is important because that is what
produces "representative" samples (i.e., samples that represent
the populations from which they were selected)!

You will see below
that, simple random samples are __not__ the only equal probability sampling
method (EPSEM). It is the most basic and well know, however.

- Sampling experts recommend random sampling "without replacement" rather than random sampling "with replacement" because the former is a little more efficient in producing representative samples (i.e., it requires slightly fewer people and is therefore a little cheaper).

“How do you draw a __simple random sample__?"

- One way is to put all the names from
your population into a hat and then select a subset (e.g., pull out 100
names from the hat).
- In
the chapter we demonstrate the use of a table of random numbers.
- These days, researchers often use computer programs to randomly select their samples.
- Here is a program the you can easily use for simple random sampling, just click here.
- To use a computer program (called a
random number generator) you must make sure that you give each of the
people in your population a number. Then the program will give you a list
of randomly selected numbers within the range you give it. After getting
the random numbers, you identify the people with those randomly selected
numbers and try to get them to participate in your research study!
- If
you decide to use a table of random numbers such as the one shown on page
201 of the book, here’s what you need to do. First, pick a place to start, and then move in
one direction (e.g., move down the columns). Use the number of digits in the table
that is appropriate for your population size (e.g., if there are 2500
people in the population then use 4 digits). Once you get the set of randomly
selected numbers, find out who those people are and try to get them to
participate in your research study. Also, if you get the same
number twice, just ignore it and move on to the next number.

**Systematic Sampling**

__Systematic sampling__ is the second type of random
sampling.

- It is an equal probability sampling
method (EPSEM).
- Remember simple random sampling was also
an EPSEM.

Systematic sampling
involves three steps:

·
First,
determine the sampling interval, which is symbolized by "k," (it is
the population size divided by the desired sample size).

·
Second,
randomly select a number between 1 and k, and include that person in your
sample.

·
Third, also
include each k^{th} element in your sample. For example if k is 10 and
your randomly selected number between 1 and 10 was 5, then you will select
persons 5, 15, 25, 35, 45, etc.

·
When you get to
the end of your sampling frame you will have all the people to be included in
your sample.

·
One potential
(but rarely occurring) problem is called __periodicity__ (i.e., there is a
cyclical pattern in the sampling frame). It could occur when you attach
several ordered lists to one another (e.g., if you had took lists from multiple
teachers who had all ordered their lists on some variable such as IQ). On the
other hand, stratification within one overall list is not a problem at all
(e.g., if you have one list and have it ordered by gender, or by IQ).
Basically, if you are attaching multiple lists to one another, there could be a
problem. It would be better to reorganize the lists into one overall list
(i.e., sampling frame).

**Stratified
Random Sampling**

The third type of random sampling is called __stratified
random sampling__.

- First, stratify your sampling frame
(e.g., divide it into the males and the females if you are using gender as
your stratification variable).
- Second, take a random sample from each
group (i.e., take a random sample of males and a random sample of
females). Put these two sets of people together and you now have your
final sample. (Note that you could also take a systematic sample
from the joined lists if that’s easier.)

There are actually two different types of stratified sampling.

The first type of
stratified sampling, and most common, is called __proportional stratified
sampling__.

- In proportional stratified sampling you
must make sure the subsamples (e.g., the samples of males and females) are
proportional to their sizes in the population.
- Note that proportional stratified
sampling is an equal probability sampling method (i.e., it is EPSEM),
which is good!

The second type of
stratified sampling is called __disproportional stratified sampling__.

·
In
disproportional stratified sampling, the subsamples are not proportional to
their

sizes in the population.

Here is an example showing the difference between proportional and disproportional stratified sampling:

- Assume
that your population is 75% female and 25% male. Assume also that you want
a sample of size 100 and you want to stratify on the variable called gender.
- For proportional stratified sampling,
you would randomly select 75 females and 25 males from the population.
- For disproportional stratified sampling,
you might randomly select 50 females and 50 males from the population.

** **

**Cluster Random Sampling**

In this type of
sampling you randomly select __clusters__ rather than individual type units
in the first stage of sampling.

- A
__cluster__has more than one unit in it (e.g., a school, a classroom, a team).

We discuss two types of cluster sampling in the chapter,
one-stage and two-stage (note that more stages are possible in multistage
sampling but are left for books on sampling).

The first type of
cluster sampling is called __one-stage cluster sampling__.

- To select a one-stage cluster sample,
you first select a random sample of clusters.
- Then you include in your final sample
all of the individual units that are in the selected clusters.

The second type of
cluster sampling is called __two-stage cluster sampling__.

- In the first stage you take a random
sample of clusters (i.e., just like you did in one-stage cluster
sampling).
- In the second stage, you take a random
sample of elements from each of the clusters you selected in stage one
(e.g., in stage two you might randomly select 10 students from each of the
15 classrooms you selected in stage one).

Important points
about cluster sampling:

- Cluster sampling is an equal probability
sampling method (EPSEM) ONLY
__if the clusters are approximately the same size__. (Remember that EPSEM is very important because that is what produces representative samples.) - When clusters are not the same size, you
must fix the problem by using the technique called "probability
proportional to size" (PPS) for selecting your clusters in stage one.
This will make your cluster sampling an equal probability sampling method
(EPSEM), and it will, therefore, produce representative samples.

**Nonrandom Sampling
Techniques**

The other major type of sampling used in quantitative research is nonrandom sampling (i.e., when you do not use one of the ransom sampling techniques). There are four main types of nonrandom sampling:

- The
first type of nonrandom sampling is called
__convenience sampling__(i.e., it simply involves using the people who are the most available or the most easily selected to be in your research study). - The
second type of nonrandom sampling is called
__quota sampling__(i.e., it involves setting quotas and then using convenience sampling to obtain those quotas). A set of quotas might be given to you as follows: find 25 African American males, 25 European American males, 25 African American females, and 25 European American females. You use convenience sampling to actually find the people, but you must make sure you have the right number of people for each quota. - The
third type of nonrandom sampling is called
__purposive sampling__(i.e., the researcher specifies the characteristics of the population of interest and then locates individuals who match those characteristics). For example, you might decide that you want to only include "boys who are in the 7th grade and have been diagnosed with ADHD" in your research study. You would then, try to find 50 students who meet your "inclusion criteria" and include them in your research study. - The
fourth type of nonrandom sampling is called
__snowball sampling__(i.e., each research participant is asked to identify other potential research participants who have a certain characteristic). You start with one or a few participants, ask them for more, find those, ask them for some, and continue until you have a sufficient sample size. This technique might be used for a hard to find population (e.g., where no sampling frame exists). For example, you might want to use snowball sampling if you wanted to do a study of people in your city who have a lot of power in the area of educational policy making (in addition to the already known positions of power, such as the school board and the school system superintendent).

**Random Selection
and Random Assignment**

In __random selection__ (using an equal
probability selection method), you select a sample from a population using one
of the random sampling techniques discussed earlier.

- The resulting random sample will be like a "mirror image" of the population, except for chance differences.
- For example, if you randomly select
(e.g., using simple random sampling) 1000 people from the adult population
in Ann Arbor, Michigan, the sample will look like the adult population of
Ann Arbor.

In
random assignment, you start with a set of people (you already have a sample,
which very well may be a convenience sample), and then you randomly divide that
set of people into two or more groups (i.e., you take the full set and randomly
divide it into subsets).

- You are taking a set of
people and “assigning” them to two or more groups.
- The
groups or subsets will be "mirror images" of each other (except
for chance differences).
- For example, if you start with a
convenience sample of 100 people and randomly assign them to two groups of
50 people, the two groups will be "equivalent" on all known and
unknown variables.
- Random assignment generates similar
groups, and it is used in the strongest of the experimental research
designs.
- To see exactly how to do random
assignment, then
click here.
- You can also use this randomizer program
for random assignment, just click here.

**Determining the
Sample Size
When Random Sampling is Used**

Would you like to know the answer to the question "How big should my sample be?"

I will start with my four "simple" answers to your question:

- Try to get as big of a sample as you can for your study (i.e., because the bigger the sample the better).
- If your population is size 100 or less, then include the whole population rather than taking a sample (i.e., don't take a sample; include the whole population).
- Look at other studies in the research literature and see how many they are selecting.
- For an exact number, just look at Figure 7.5 which shows recommended sample sizes.
- There are many sample size calculators on the web but they generally require you to learn a little bit of statistics first. Here is one click here. I’ll list more when we get to the chapter on statistics.

I want to make a few more points about sample size in this
chapter. In particular, note that you will need __larger__ samples under
these circumstances:

- When
the population is very
__heterogeneous__. - When you want to breakdown the data into multiple categories.
- When
you want a relatively
__narrow confidence interval__(e.g., note that the estimate that 75% of teachers support a policy plus or minus 4% is more narrow than the estimate of 75% plus or minus 5%). - When
you expect a
__weak relationship__or a__small effect__. - When
you use a
__less efficient technique__of random sampling (e.g., cluster sampling is less efficient than proportional stratified sampling). - When
you expect to have a
__low response rate.__The response rate is the percentage of people in your sample who agree to be in your study.

**Sampling in
Qualitative Research**

Sampling in __qualitative__ research is usually __purposive
__(see the above discussion of purposive sampling). The primary goal in
qualitative research is to select information rich cases.

There are several specific purposive sampling techniques that are used in qualitative research:

__Maximum variation sampling__(i.e., you select a wide range of cases).__Homogeneous sample selection__(i.e., you select a small and homogeneous case or set of cases for intensive study).__Extreme case sampling__(i.e., you select cases that represent the extremes on some dimension).__Typical-case sampling__(i.e., you select typical or average cases).__Critical-case sampling__(i.e., you select cases that are known to be very important).__Negative-case sampling__(i.e., you purposively select cases that disconfirm your generalizations, so that you can make sure that you are not just selectively finding cases to support your personal theory).__Opportunistic sampling__(i.e., you select useful cases as the opportunity arises).__Mixed purposeful sampling__(i.e., you can mix the sampling strategies we have discussed into more complex designs tailored to your specific needs).

For a little more information on sampling in qualitative research, click here. (Hit the right arrow key to move from slide to slide.)