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Chapter 6
Sampling Distributions
The Concept of Sampling
Distributions
Parameter – numerical descriptive measure
of a population. It is usually unknown
Sample Statistic - numerical descriptive
measure of a sample. It is usually known
Sampling distribution – the probability
distribution of a sample statistic, calculated
from a very large number of samples of size
n
The Concept of Sampling
Distributions
We can take 45 samples of size 2 from this
group of 10 observations
If we take one random sample and get (19,
Another random sample may yield (22, 25),
with
x = 19. 5
x = 23. 5
The Concept of Sampling
Distributions
Taking all possible samples of size 2, we
can graph them and come up with a
sampling distribution of the sample statistic
Sampling distributions can be derived for
x
Sampling distributions can be derived for
any statistic
Knowing the properties of the underlying
sampling distributions allows us to judge
how accurate the statistics are as estimates
of parameters
The Concept of Sampling
Distributions
Decisions about which sample statistic to
use must take into account the sampling
distribution of the statistics you will be
h i f
choosing from.
The Concept of Sampling
Distributions
Given the probability distribution
X 0 6 9
Find the sampling distribution of mean and
median of x
p(x) 1/3 1/3 1/
Possible Samples x m Probability
0,0,0 0 0 1/ 0,0,6 2 0 1/ 0,0,9 3 0 1/ 0,6,0 2 0 1/ 0,6,6 4 6 1/ 0,6,9 5 6 1/ 0,9,0 3 0 1/ 0,9,6 5 6 1/ 0,9,9 6 9 1/ 6,0,0 2 0 1/ 6,0,6 4 6 1/ 6,0,9 5 6 1/ 6,6,0 4 6 1/ 6 6 6 6 6 1/
x P(x)
Sampling distribution
of x
m P(m)
Sampling distribution
of m
The Concept of Sampling
Distributions
Simulating a Sampling Distribution
Use a software package to generate samples of size n = 11
from a population with a known μ =.
Calculate the mean and median for each sample
p
Generate histograms for the means and medians of the
samples
Note the greater clustering of
the values of around μ
These histograms are
approximations of the sampling
distributions of and m
x
x
Properties of Sampling Distributions:
Unbiasedness and Minimum Variance
Point Estimator – formula or rule for using sample
data to calculate an estimate of a population
parameter
Point estimators have sampling distributions
Point estimators have sampling distributions
These sampling distributions tell us how accurate
an estimate the point estimator is likely to be
Sampling distributions can also indicate whether
an estimator is likely to under/over estimate a
parameter
Properties of Sampling Distributions:
Unbiasedness and Minimum Variance
Two point estimators, A and B, of parameter θ
After generating the sampling distributions of A and B, we
can see that
A is an unbiased estimator of θ
B is a biased estimator of θ, with a bias toward
overstatement
Properties of Sampling Distributions:
Unbiasedness and Minimum Variance
What if A and B are both unbiased estimators of θ?
Look at the sampling distributions and compare their
standard deviations
A has a smaller standard
deviation than B
Which would you use as
your estimator?
The Sampling Distribution of X and
the Central Limit Theorem
Assume 1000 samples of size n taken from a population,
with calculated for each sample. What are the Properties
of the Sampling Distribution of?
Mean of sampling distribution equals mean of sampled
x
x
( )
population
Standard deviation of sampling distribution equals
Standard deviation of sampled population
Square root of sample size
or,
is referred to as the standard error of the mean
u = E ( x )= μ x
n
x
σ = σ
x
σ
