Slides to accompany Weathington, Cunningham & Pittenger (2010),

Chapter 7: Sampling

1

Objectives

• Samples, in general

• Probability sampling

• Probability sampling methods

• Nonprobability sampling

• Central Limit Theorem

• Applications of CLT

• Sources of bias and error

2

Why Worry about Sampling?

• Don’t worry, just appreciate it

• Objective sampling helps us avoid the Idols of the Cave

– Improving external validity of our conclusions

• “Good” sampling allows us to make comparisons and predictions from our data

3

Samples...

• …are (hopefully) valid representatives of the population you are studying

• …can grant you better (more objective, empirical) data than you will find in anecdotes

• …allow you to avoid reliance on one person’s opinions, perspectives, and biases

4

Probability Examples

• Probability of Heads in one flip of a fair coin:

p(H) = 1/2; p(T)=1/2=.5

• p(H and T) in two flips = 2/4=.5

• p(correct answer on 4-option mc question) = .25

• Pr. of choosing a woman in a single random selection from a class of 223 students with 150 women: p(w)=150/223=.673

5

Probability Sampling

• Random: each outcome has an equal probability of occurring, every time

– Every time I flip a coin, the probability is .5 that it will be H or T

• Random sampling depends on this independence of outcomes

• Law of large numbers: On average, a large selection of items will have the same characteristics as those in the population

6

Populations and Samples

• Target vs. sampling population

– Target: (universe) e.g. all depressed persons

– Sampling: (accessible) all diagnosed as depressed

• Sampling Frame (all who can be reached)

• Subject (participant pool) – (willing to participate)

• Descriptive data helps us compare our sample against the population

• External validity depends largely on representativeness in sampling

7

Probability Sampling Characteristics• Each population member has an equal

chance of being a potential sample member

– No systematic exclusions

• Sampling procedures are based on a protocol

– Prevents bias effects on sample selection

• Probability of any specific sample can be calculated

– Helps connect results with population8

Simple Random Sampling

• Each population member has equal probability of selection to the sample

– If selection is random, the sample of any size should represent the population from which it was chosen

• Random numbers are in tables and Excel-type computer programs

9

Simple Random Sampling: How-To

• Generate a list of possible participants (population) in Microsoft Excel

• In the next column insert the function “=RAND()”

– Creates a random number between 0 and 1

• Sort both columns by the random numbers

• Select the first N individuals for your sample

10

Sequential/Systematic Sampling• Random is not always practical

• All sampling population members are listed and each kth member is selected to the sample

k = sampling interval = Population size

desired sample N

11

Stratified Sampling

• Good option when sample needs to include subgroups from a population

– Based on gender, age, education, etc.

• Size of subgroups in final sample must be equivalent to size in population

• Can use simple random or sequential sampling to fill each relative subgroup

12

Cluster Sampling

• Good option when participants are already in groups that cannot be easily separated

– e.g., Study of coaching’s impact on different sports teams

• Instead of randomly selecting team members, you randomly select teams

• If need certain subgroup representation, this may limit your option of teams

13

Nonprobability Sampling

• Sampling based on some other factor besides probability

– May be more convenient

– May not be as representative

•Can’t establish probabilities associated with sample membership

– Can still be useful if treated with caution

14

Convenience Sampling

• “Person” on the street approach

• Sampling from easy to find population members (a “special” subset)

• Sample determined in part by researcher’s sampling method

– Not by probability

• Can bias/distort results

• Sometimes the only option

15

Snowball Sampling

• Good for cohort studies or when trying to reach a dispersed population

• Using one cohort member to find others, and so on...

• Pros: Good for research on difficult populations to reach (e.g., homeless)

• Cons: No representative sample guarantee

16

Central Limit Theorem

Refers to distribution of characteristics within the probability samples

1. As N (sample size) increases, the shape of the sampling distribution of means will approach a normal distribution

2. µM = µ (mean of sample means =pop

mean)

3. σM = σ/√n (SEM)

17

CLT • Sampling Distribution Shape

– Figure 7.4 Note how the M becomes closer to µ as N increases

• µM = mean of means = (sum of all sample

means)/(number of samples)

– M = unbiased estimate of µ

• σM = std. dev. of the sampling distribution of M

– As n increases, distribution of sample means will cluster closer to µ more accurate estimate

18

19

CLT

• If we use probability sampling, M = unbiased estimate of µ

• M becomes a better estimate of µ when n increases

• We can determine the probability of obtaining various M

20

Standard Error of the Mean

• Represents uncertainty of how well M represents µ

• SEM = SD of sampling distribution of means

σ / √n (n = sample size)

http://www.miniwebtool.com/standard-error-calculator/

• SEM is affected by:

– σ as this decreases, SEM decreases

–n as this increases, SEM decreases (1/√n)

• M is best estimate of µ when SEM is low21

Applying CLT

• Reliability of a sample mean (M)

– Use SEM to calculate confidence intervals around M (see Fig 7.4, p 212)

– There will be variability among sample M, but a CI can help you determine the expected range

• Adequacy of a sample size (n)

22

Confidence Intervals

• In a normal distribution, 68% of M within 1 SEM of µ, 95% within 1.96 SEM of, 99% within 2.58 SEM

• Can use CI to predict other M

– 95% CI = 95% of future sample M should fall within this range

23

Sources of Bias and Error

• Bias: nonrandom, systematic factors that may make M differ from µ

– Could be controlled

• Error: random events that have the same effect, but cannot be controlled

• Figure 7.7 is a good illustration

– Ideally, µ’ = µ, but not in these examples

– Possible nonsampling biases at work

24

25

Bias and Error

• If the sampling is random, then even if there is a nonsampling bias present, µM

= µ’

• Sampling bias: systematic selection bias while sampling

• Total error = M - µ

– Sum of effects from nonsampling bias, sampling bias, and sampling error

26

What is Next?

• **instructor to provide details

27