Transcript Sampling Distributions and Forward Inference • By simulating the process of drawing random samples of size N from a population with a.

Sampling Distributions and Forward Inference
• By simulating the process of drawing random samples of
size N from a population with a specific mean and
variance, we can learn
– (a) how much error we can expect on average and
– (b) how much variation there will be on average in the
errors observed
• Sampling distribution: the distribution of a sample
statistic (e.g., a mean) when sampled under known
sampling conditions from a known population.
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z
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n=2
n=5
n = 15
mean of sample
means = 10
mean of sample
means = 10
mean of sample
means = 10
SD of sample means =
4.16
SD of sample means =
2.41
SD of sample means =
0.87
The mean as an unbiased statistic
• Note that the distributions of sample means were normal
and centered at the mean of the population.
• Thus, we “expect” or predict any sample mean to equal the
population mean. Why? The average (i.e., typical) sample
mean is equal to the mean of the population.
• In this sense, the mean is considered an “unbiased”
statistic. To the degree that a sample mean differs from a
population mean, it is just as likely to be too high or too
low.
expectedM x   
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expecteds    1  
N
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Sample variance
• The value of the sample
variance we “expect,”
however, is not equal to the
population variance.
• Specifically, a variance
observed in a sample drawn
from a population will tend to
be smaller than the population
variance, especially when
sample sizes are small.
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The variance as a biased statistic
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Sample s iz e (N )
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Standard Error of the Mean
• The amount of sampling error we expect is an average
error--the standard deviation of the sampling distribution.
This represents, on average, how much of an error we
should expect.
• The standard deviation of a sampling distribution is often
called a standard error.
Standard Error of the Mean
• When dealing with means, we call this the standard error
of the mean.
• Its calculation is simple:
SEM x 
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or
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To find the SD of the sampling distribution of
means, all you need to know is the standard
deviation of scores in the population, and your
sample size. [in forward inference]
Standard Error of the Mean
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SEM
• The equation implies that
sampling error decreases as
sample size increases.
• This is important because it
implies that if we want to
make sampling error as
small as possible, we want
to use as large of a sample
size as we can manage.
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SEM x 
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