Chapter 10:
Sampling and Sampling Distributions
•
•
•
•
•
•
•
Aims of Sampling
Basic Principles of Probability
Types of Random Samples
Sampling Distributions
Sampling Distribution of the Mean
Standard Error of the Mean
The Central Limit Theorem
Chapter 11 – 1
Sampling
• Population – A group that includes all the
cases (individuals, objects, or groups) in
which the researcher is interested.
• Sample – A relatively small subset from a
population.
Chapter 11 – 2
Notation
Chapter 11 – 3
Sampling
• Parameter – A measure (for example, mean
or standard deviation) used to describe a
population distribution.
• Statistic – A measure (for example, mean or
standard deviation) used to describe a
sample distribution.
Chapter 11 – 4
Sampling: Parameter & Statistic
Chapter 11 – 5
Probability Sampling
• Probability sampling – A method of sampling that
enables the researcher to specify for each case in
the population the probability of its inclusion in the
sample.
Chapter 11 – 6
Random Sampling
• Simple Random Sample – A sample
designed in such a way as to ensure that (1)
every member of the population has an
equal chance of being chosen and (2) every
combination of N members has an equal
chance of being chosen.
• This can be done using a computer,
calculator, or a table of random numbers
Chapter 11 – 7
Population inferences can be made...
Chapter 11 – 8
...by selecting a representative sample
from the population
Chapter 11 – 9
Random Sampling
• Systematic random sampling – A method of
sampling in which every Kth member (K is a
ration obtained by dividing the population size
by the desired sample size) in the total
population is chosen for inclusion in the
sample after the first member of the sample is
selected at random from among the first K
members of the population.
Chapter 11 – 10
Systematic Random Sampling
Chapter 11 – 11
Stratified Random Sampling
• Stratified random sample – A method of
sampling obtained by (1) dividing the
population into subgroups based on one or
more variables central to our analysis and
(2) then drawing a simple random sample
from each of the subgroups
Chapter 11 – 12
Stratified Random Sampling
• Proportionate stratified sample – The size
of the sample selected from each subgroup
is proportional to the size of that subgroup
in the entire population.
• Disproportionate stratified sample – The
size of the sample selected from each
subgroup is disproportional to the size of
that subgroup in the population.
Chapter 11 – 13
Disproportionate Stratified Sample
Chapter 11 – 14
Sampling Distributions
• Sampling error – The discrepancy between
a sample estimate of a population parameter
and the real population parameter.
• Sampling distribution – A theoretical
distribution of all possible sample values for
the statistic in which we are interested.
Chapter 11 – 15
Sampling Distributions
• Sampling distribution of the mean – A theoretical
probability distribution of sample means that would
be obtained by drawing from the population all
possible samples of the same size.
If we repeatedly drew samples from a population and
calculated the sample means, those sample means would be
normally distributed (as the number of samples drawn
increases.) The next several slides demonstrate this.
• Standard error of the mean – The standard
deviation of the sampling distribution of the mean.
It describes how much dispersion there is in the
sampling distribution of the mean.
Chapter 11 – 16
Sampling
Distributions
Chapter 11 – 17
Distribution of Sample Means
with 21 Samples
10
S.D. = 2.02
Mean of means = 41.0
Number of Means = 21
Frequency
8
6
4
2
0
37
38
39
40
41
42
43
44
45
46
Sample Means
Chapter 11 – 18
Distribution of Sample Means
with 96 Samples
14
S.D. = 1.80
Mean of Means = 41.12
Number of Means = 96
12
Frequency
10
8
6
4
2
0
37
38
39
40
41
42
43
44
45
46
Sample Means
Chapter 11 – 19
Distribution of Sample Means
with 170 Samples
30
S.D. = 1.71
Mean of Means= 41.12
Number of Means= 170
Frequency
20
10
0
37
38
39
40
41
42
43
44
45
46
Sample Means
Chapter 11 – 20
The Central Limit Theorem
• If all possible random samples of size N are
drawn from a population with mean y and
a standard deviation  y , then as N becomes
larger, the sampling distribution of sample
means becomes approximately normal, with
mean y and standard deviation  y / N .
Chapter 11 – 21