Let’s use the means we obtained
from these samples of n =2 and
construct a frequency distribution of
means. This is called the sampling
distribution of the mean:
Sampling Distributions
Sampling distribution: Theoretical distribution that shows the frequency of each possible value of the
statistic from sample size n when
1)
the null hypothesis is true and
2)
the population has been sampled randomly, with replacement, until all possible samples have
been taken.
Example 1. Population with only five scores: 3, 4, 5, 2, 6
(
μ
= 4). Sketch this distribution below using a histogram.
Now let’s randomly select samples of 2 scores (n = 2) from this population until all possible samples have been
taken. The two scores listed under sample scores are the samples of size 2. A mean is computed for each sample.
Sample
Scores
M
M
f
3+3
3
6
1
3+4
3.5
5.5
2
3+5
4
5
3
3+2
2.5
4.5
4
3+6
4.5
4
5
4+3
3.5
3.5
4
4+4
4
3
3
4+5
4.5
2.5
2
4+2
3
2
1
4+6
5
25 = N
5+3
4
Notice:
5+4
4.5
μ
M
=
4
μ
M
=
mean of the sample means
5+2
3.5
σ
M
=
1.02
σ
M
=
std. dev. of the sample means
5+5
5
5+6
5.5
Sketch this distribution of means:
2+2
2
2+3
2.5
2+4
3
2+5
3.5
2+6
4
6+3
4.5
6+4
5
6+5
5.5
6+6
6
6+2
4
Dr. Viji Sathy, Psychology 210 Class Notes, 1
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Sampling distributions are used to determine probability.
•
If we select two scores at random from the population, what is the probability that this sample will have a
mean of 3.5 to 4.5?
•
If we select two scores at random from the population, what is the probability that this sample will have a
mean of 5.5 or greater?
Important Concepts Related to Sampling Distributions
1. Expected value: Mean of the sample statistics; mean of the sampling distribution.
•
If a statistic is unbiased, then the expected value = value of the parameter
•
For the sampling distribution of the mean:
μ
M
=
μ
2. Sampling error: Difference between the sample statistic and the population parameter
3. Standard error: Average difference between sample statistic and population parameter
•
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 Fall '08
 Chow
 Dr. Viji Sathy

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