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Sampling Distributions
Terms
 Parameter
- a number (usually unknown) that
describes a population.
 Statistic – a number that can be computed from
sample data. A statistic is used to estimate an
unknown parameter.
 Sampling Variability – Due to random chance, results
from different samples will vary.
 *Remember* - There is always error in sampling.
Example
 Suppose that 25% of all US adults are either single or
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divorced.
.25 is a parameter.
We take a random sample of 50 US adults and
determine the proportion of them who are divorced or
single. We repeat this 5 times. Our results are:
.29, .32, .21, .24, .28. These values are statistics.
Note that these values vary around the true value.
How to Handle Sampling Variability
 Take a large number of samples of the same size.
 Calculate the desired statistic for each sample.
 Make a histogram of the values of the statistic.
 Examine the histogram for CUSS.
 If one were able to do this an infinite amount of times,
the histogram would approach a smooth curve
containing all possible samples of that size. This curve
is known as a Sampling Distribution.
Definition
 The Sampling Distribution of a statistic is the
probability distribution of values (means, proportions,
etc.) taken by the statistic in all possible samples of the
same size from the same population.
 A Sampling Distribution may be comprised of means,
proportions, slopes, etc. We will start with means.
 They are described using CUSS.
 For means, we will speak of the “sampling distribution
of x-bar”.
Bias of a Statistic
 The bias of a statistic concerns the center of the
distribution.
 A statistic is unbiased if:  x  
 That is, if the sampling distribution mean is equal to
the true mean of the parameter, for all possible samples
of a given size.
Variability of a Statistic
 The variability of a statistic is described by the spread
of the sampling distribution.
 The spread is determined by the sampling design and
by the size of the sample.
 Larger samples produce less variation (less error in
sampling).
 For identical sample sizes, as long as the population is
at least 10 times the sample size, variability won’t be
affected by the size of the population.
How Bias and Variability Interact
Homework
 Textbook: 9.1-9.4, 9.10-9.13