This document discusses sampling and sampling distributions. It defines key concepts like population, sample, probability distributions, sampling distributions, and the central limit theorem. It explains that as sample size increases, the sampling distribution approximates a normal distribution according to the central limit theorem. It also discusses different types of sampling methods like simple random sampling, systematic random sampling, and stratified random sampling.

1. Sampling and Sampling Distributions
 Aims of Sampling
 Probability Distributions
 Sampling Distributions
 The Central Limit Theorem
 Types of Samples

2. Aims of sampling
 Reduces cost of research (e.g. political
polls)
 Generalize about a larger population (e.g.,
benefits of sampling city r/t neighborhood)
 In some cases (e.g. industrial production)
analysis may be destructive, so sampling
is needed

3. Probability
 Probability: what is the chance that a
given event will occur?
 Probability is expressed in numbers
between 0 and 1. Probability = 0 means
the event never happens; probability = 1
means it always happens.
 The total probability of all possible event
always sums to 1.

4. Probability distributions: Permutations
What is the probability distribution of number
of girls in families with two children?
2 GG
1 BG
1 GB
0 BB

5. Probability Distribution of
Number of Girls
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2

6. How about family of three?
Num. Girls child #1 child #2 child #3
0 B B B
1 B B G
1 B G B
1 G B B
2 B G G
2 G B G
2 G G B
3 G G G

7. Probability distribution of number of girls
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3

8. How about a family of 10?
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10

9. As family size increases, the binomial
distribution looks more and more normal.
Number of Successes
3.02.01.00.0
Number of Successes
10987654321-0

10. Normal distribution
Same shape, if you adjusted the scales
CA
B

11. Coin toss
 Toss a coin 30 times
 Tabulate results

12. Coin toss
 Suppose this were 12 randomly selected
families, and heads were girls
 If you did it enough times distribution would
approximate “Normal” distribution
 Think of the coin tosses as samples of all
possible coin tosses

13. Sampling distribution
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.

14. Central Limit Theorem
 No matter what we are measuring, the
distribution of any measure across all
possible samples we could take
approximates a normal distribution, as
long as the number of cases in each
sample is about 30 or larger.

15. Central Limit Theorem
If we repeatedly drew samples from a
population and calculated the mean of a
variable or a percentage or, those sample
means or percentages would be normally
distributed.

16. Most empirical distributions are not normal:
U.S. Income distribution 1992

17. But the sampling distribution of mean income over
many samples is normal
Sampling Distribution of Income, 1992 (thousands)
18 19 20 21 22 23 24 25 26
N
u
m
b
e
r
o
f
s
a
m
p
l
e
s
Numberofsamples

18. Standard Deviation
Measures how spread
out a distribution is.
Square root of the sum
of the squared
deviations of each
case from the mean
over the number of
cases, or
( )
N
Xi∑ −
=
2
µ
σ

19. Deviation from Mean
Amount X (X - X) ( X - X )
600 435 600 - 435 = 165 27,225
350 435 350 - 435 = -85 7,225
275 435 275 - 435 = -160 25,600
430 435 430 -435 = -5 25
520 435 520 - 435 = 85 7,225
0 67,300
( )X X
n
−
−
∑
1
s = = = = 129.7167 300
4
,
16 825,
2
2
Example of Standard Deviation

20. Standard Deviation and Normal Distribution

21. 10
8
6
4
2
0
37 38 39 40 41 42 43 44 45 46
Sample Means
S.D. = 2.02
Mean of means = 41.0
Number of Means = 21
Distribution of Sample Means with 21
Samples
Frequency

22. Frequency
14
12
10
8
6
4
2
0 37 38 39 40 41 42 43 44 45 46
Sample Means
Distribution of Sample Means with 96
Samples
S.D. = 1.80
Mean of Means = 41.12
Number of Means = 96

23. Distribution of Sample Means with 170
Samples
Frequency
30
20
10
0 37 38 39 40 41 42 43 44 45 46
Sample Means
S.D. = 1.71
Mean of Means= 41.12
Number of Means= 170

24. The standard deviation of the sampling
distribution is called the standard error

25. Standard error can be estimated from a single sample:
The Central Limit Theorem
Where
s is the sample standard deviation (i.e., the
sample based estimate of the standard deviation of the
population), and
n is the size (number of observations) of the sample.

26. Because we know that the sampling distribution is normal, we
know that 95.45% of samples will fall within two standard errors.
95% of samples fall within 1.96
standard errors.
99% of samples fall within
2.58 standard errors.
Confidence intervals

27. 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.

28. 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

29. Population inferences can be made...

30. ...by selecting a representative sample from
the population

31. 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.

32. Systematic Random Sampling

33. 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. (Self weighting)
 Disproportionate stratified sample – The
size of the sample selected from each
subgroup is disproportional to the size of that
subgroup in the population. (needs weights)

34. Disproportionate Stratified Sample

35. 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