Determining the Sampling Plan
Audhesh Paswan, Ph.D.
University of North Texas
Audhesh Paswan, Ph.D.
Basic Concepts in Samples
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Population - the entire group under study as
specified by the objectives of the research
project.
Sample - a subset of the population that
should represent that entire group.
Sample unit - the basic level of investigation
(e.g., household, individual).
Census - an accounting of the complete
population.
Audhesh Paswan, Ph.D.
Basic Concepts in Sampling
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Sampling error - error in a survey that
occurs because a sample is used (caused by
two factors).
• The method of sample selection
• The size of the sample
Sample frame - a list of the population of
interest.
Sample frame error - the degree to which a
sample frame fails to account for all of the
population.
Audhesh Paswan, Ph.D.
Examples of sample frame errors
Phone book
 Yellow pages
 Any incomplete population lists

Audhesh Paswan, Ph.D.
Why sample?
A sample is almost always more desirable
than a census
 Population size and expense.
 Cannot analyze the huge amount of
data generated by a census.
Audhesh Paswan, Ph.D.
Basic Sampling Methods
Probability samples - members of the
population have a known chance (I.e.,
probability) of being selected into the
sample.
 Nonprobability samples - the chances
(I.e., probabilities) of selecting members
from the population of interest into the
sample are unknown.

Audhesh Paswan, Ph.D.
Probability Methods - Simple
Random Sampling

Probability is known and equal for all
members of the population.
P(selection)=(sample size)/(population
size)
The "Blind Draw" Method
 The Table of Random Numbers
Method

Audhesh Paswan, Ph.D.
Advantages and Disadvantages

Advantages
• Derives unbiased estimates
• Valid representation of the population

Disadvantages
• Must pre-designate each population
member.
• May be difficult to obtain a complete listing.
• May be too cumbersome
Audhesh Paswan, Ph.D.
Systematic Sampling

One of most prevalent types used

Advantage: "economic efficiency" (i.e.,
quick and easy).

It employs a random starting point

Every kth element in the population is
designated for inclusion in the sample (after a
random start).

Create a sample that is almost identical in
quality to simple random sampling.
Audhesh Paswan, Ph.D.
Systematic Sample
Example 1: Sample the population of
phone customers in Denton by taking
every 10th number in the phone book.
(Be sure to start randomly on one of the
first 10 numbers.)
 Example 2: Sample every 5th customer

Skip interval = (population list size)/(sample
size)
Audhesh Paswan, Ph.D.
How to Take a Systematic Sample
Step 1: Identify a listing of the population
that contains an acceptable level of
sample frame error.
Step 2: Compute the skip interval.
Step 3: Using random number(s),
determine a starting position.
Step 4: Apply the skip interval.
Step 5: Treat the list as "circular."
Audhesh Paswan, Ph.D.
Cluster Sampling
Population is divided into subgroups
(cluster)
 Each cluster represents the entire
population.
 Must identify clusters that are identical
to the population and to each other.
 The parent population is divided into
mutually exclusive and exhaustive
subsets.

Audhesh Paswan, Ph.D.
Cluster Sampling

Subgroups should be heterogeneous within
and homogeneous between. (i.e., Subsets
should each look representative of the total
population.)

Advantages: less cost to obtain a sample;
good for personal interviews (proximity)

Limitations: difficult to find subsets that truly
meet the criteria mentioned above; lower
statistical efficiency (higher error)
Audhesh Paswan, Ph.D.
Area Sampling as a Form of
Cluster Sampling

Population subdivided into areas (e.g., cities or
neighborhoods)
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One-step approach - one area is selected
randomly; perform a census of the cluster
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Two-step approach
• Step 1: Select a random sample of clusters.
• Step 2: Randomly select individuals within the
clusters.
Audhesh Paswan, Ph.D.
How to Take an Area Sampling
Using Subdivisions
Step 1: Determine the geographic area to be
surveyed, and identify its subdivisions. Each
subdivision should be highly similar to all others.
Step 2: Decide on the use of one-step or two-step
cluster sampling.
Step 3: Using random numbers, select the
subdivisions to be sampled (Assuming two-step).
Step 4: Using some probability method of sample
selection, select the members of each chosen
subdivision to be included in the sample.
Audhesh Paswan, Ph.D.
Stratified Sampling
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Separates the population into different
subgroups and then samples all of the
subgroups.
Does not assume the population has a
"normal" distribution.
Addresses "skewed" distribution problems.
Weighted mean
Meanpopulation = (meanA)(proportionA)+
(meanB)(proportionB)
Audhesh Paswan, Ph.D.
Stratified Sample

A probability sample distinguished by a
two-step procedure:
1 Divide the population into mutually
exclusive and collectively exhaustive
subsets.
2 Take a simple random sample of elements
from each subset (independently).
The subsets are called “strata”.
 Each population member can be
assigned to one and only one stratum.

Audhesh Paswan, Ph.D.
Stratified Sampling
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Advantages:
• Produces a more concentrated distribution of
estimates (leads to more precise statistics and
smaller sampling error); fewer possible sample
means that deviate widely from the true population
mean.
• Can reduce variation within each stratum, which
reduces the error of the estimate
• Guarantees representation of certain subgroups of
interest.
Limitation: cost of sampling several strata
Audhesh Paswan, Ph.D.
Bases for Stratification
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Strata should be divided by a known
characteristic that is expected to be related to
the characteristic of interest.
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Example: If we are interested in magazine
readership, we can stratify on the basis of
education level. This should result in less
variation within each stratum.
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The strata should be homogeneous within
and heterogeneous between groups.
Audhesh Paswan, Ph.D.
How to Take a Stratified Sample
Step 1: Be assured that the population's distribution for some key
factor is not bell-shaped and that separate populations exist.
Step 2: Use this factor or some surrogate variable to divide the
population into strata consistent with the separate subpopulations identified.
Step 3: Select a probability sample from each stratum
Step 4: Examine each stratum for managerially relevant
differences.
Step 5: If strata sample sizes are not proportionate to the stratum
sizes in the population, use the weighted mean formula to
estimate the population value(s).
Audhesh Paswan, Ph.D.
Nonprobability Samples
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Convenience samples - drawn at the
convenience of the interviewer.
Judgement samples - requires and "educated
guess" as to who should represent the population
Referral samples - a.k.a. "snowball samples"
Quota samples - a specified quota for various
types of individuals to be interviewed is
established.
Audhesh Paswan, Ph.D.
Stratified versus Quota Sample
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Similarities:
• Population is divided into segments (strata).
• Elements are selected from each segment.
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Key Difference:
• Stratified sampling uses probability methods.
• Quota samples are based on a researcher’s
judgment.
• Therefore, stratified sampling allows the
establishment of the sampling distribution,
confidence intervals and statistical tests.
Audhesh Paswan, Ph.D.
Developing a Sample Plan
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7:
Define the relevant population
Obtain sample frame
Design the sample plan
Access the population
Draw the sample
Validate the sample
Resample, if necessary.
Audhesh Paswan, Ph.D.
Determining Sample Size
Audhesh Paswan, Ph.D.
Sample Accuracy
How close the sample’s profile is to the
true population’s profile
 Sample size is not related to
representativeness,
 Sample size is related to accuracy

Audhesh Paswan, Ph.D.
Methods of Determining Sample Size
Compromise between what is
theoretically perfect and what is
practically feasible.
 Remember, the larger the sample size,
the more costly the research.
 Why sample one more person than
necessary?

Audhesh Paswan, Ph.D.
Methods of Determining Sample Size
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Arbitrary
• Rule of Thumb (ex. A sample should be at
least 5% of the population to be accurate
• Not efficient or economical
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Conventional
• Follows that there is some “convention” or
number believed to be the right size
• Easy to apply, but can end up with too
small or too large of a sample
Audhesh Paswan, Ph.D.
Methods of Determining Sample Size
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Cost Basis
• based on budgetary constraints
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Statistical Analysis
• certain statistical techniques require certain
number of respondents
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Confidence Interval
• theoretically the most correct method
Audhesh Paswan, Ph.D.
Notion of Variability
Little
variability
Great
variability
Mean
Audhesh Paswan, Ph.D.
Notion of Variability
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Standard Deviation
• approximates the average distance away
from the mean for all respondents to a
specific question
• indicates amount of variability in sample
• ex. compare a standard deviation of 500
and 1000, which exhibits more variability?
Audhesh Paswan, Ph.D.
Measures of Variability
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Standard Deviation: indicates the degree of variation
or diversity in the values in such as way as to be
translatable into a normal curve distribution
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Variance = (x-x)2/ (n-1)
With a normal
curve, the midpoint (apex) of
i
the curve is also the mean and exactly 50%
of the distribution lies on either side of the
mean.

Audhesh Paswan, Ph.D.
Normal Curve and Standard
Deviation
Number of
standard
deviations
from the
mean
+/- 1.00 st dev
Percent of
area under
the curve
Percent of
area to the
right or left
68%
16%
+/- 1.64 st dev
90%
5%
+/- 1.96 st dev
95%
2.5%
+/- 2.58 st dev
99%
0.5%
Audhesh Paswan, Ph.D.
Notion of Sampling Distribution
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The sampling distribution refers to what
would be found if the researcher could take
many, many independent samples
The means for all of the samples should align
themselves in a normal bell-shaped curve
Therefore, it is a high probability that any
given sample result will be close to but not
exactly to the population mean.
Audhesh Paswan, Ph.D.
Normal, bell-shaped curve
Midpoint
(mean)
Audhesh Paswan, Ph.D.
Notion of Confidence Interval
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A confidence interval defines endpoints
based on knowledge of the area under a bellshaped curve.
Normal curve
• 1.96 times the standard deviation theoretically
defines 95% of the population
• 2.58 times the standard deviation theoretically
defines 99% of the population
Audhesh Paswan, Ph.D.
Notion of Confidence Interval
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Example
• Mean = 12,000 miles
• Standard Deviation = 3000 miles
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We are confident that 95% of the
respondents’ answers fall between
6,120 and 17,880 miles
12,000 + (1.96 * 3000) = 17,880
12,000 - (1.96 * 3000) = 6.120
Audhesh Paswan, Ph.D.
Notion of Standard Error of a Mean
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Standard error is an indication of how far away
from the true population value a typical sample
result is expected to fall.
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Formula
• S X = s / (square root of n)
• S p = Square root of {(p*q)/ n}
–
–
–
–
–
where S p is the standard error of the percentage
p = % found in the sample and q = (100-p)
S X is the standard error of the mean
s = standard deviation of the sample
n = sample size
Audhesh Paswan, Ph.D.
Computing Sample Size Using The
Confidence Interval Approach
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To compute sample size, three factors
need to be considered:
• amount of variability believed to be in the
population
• desired accuracy
• level of confidence required in your
estimates of the population values
Audhesh Paswan, Ph.D.
Determining Sample Size Using a
Mean
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Formula: n = (pqz2)/e2
Formula: n = (s2z2)/e2
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Where
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• n = sample size
• z = level of confidence (indicated by the number of
standard errors associated with it)
• s = variability indicated by an estimated standard
deviation
• p = estimated variability in the population
• q = (100-p)
• e = acceptable error in the sample estimate of the
population
Audhesh Paswan, Ph.D.
Determining Sample Size Using a
Mean: An Example
95% level of confidence (1.96)
 Standard deviation of 100 (from
previous studies)
 Desired precision is 10 (+ or -)
 Therefore n = 384
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• (1002 * 1.962) / 102
Audhesh Paswan, Ph.D.
Practical Considerations in
Sample Size Determination
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How to estimate variability in the
population
• prior research
• experience
• intuition

How to determine amount of precision
desired
• small samples are less accurate
• how much error can you live with?
Audhesh Paswan, Ph.D.
Practical Considerations in
Sample Size Determination

How to calculate the level of confidence
desired
• risk
• normally use either 95% or 99%
Audhesh Paswan, Ph.D.
Determining Sample Size
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Higher n (sample size) needed when:
• the standard error of the estimate is
high (population has more variability in
the sampling distribution of the test
statistic)
• higher precision (low degree of error) is
needed (i.e., it is important to have a
very precise estimate)
• higher level of confidence is required

Constraints: cost and access
Audhesh Paswan, Ph.D.
Notes About Sample Size
Population size does not determine
sample size.
 What most directly affects sample size
is the variability of the characteristic in
the population.

• Example: if all population elements have
the same value of a characteristic, then
we only need a sample of one!
Audhesh Paswan, Ph.D.