Monte Carlo: Simulating Sampling Distributions
This very nice applet is now blocked by both browsers and Java. Even if you modify Java,
adding this site to the exception list, it still blocks it. Java no longer allows you to lower security to
medium. Java does not want you to learn about sampling distributions.
I prefer to do this with SAS, but there is a
useful applet at the Online Psychology
Laboratory. that makes it much easier to do (but
gives you less control). Go there, read the
instructions, and then click on Begin.
You may find that your computer blocks
the Java applet from running. If so, you must
change your security settings. Go to the Control
Panel and select Java. Change the security
setting to Medium, Apply, OK. After you have
finished with the app, set the security back to
High or Very High.
Your browser may give you a security warning.
Click Run.
First, obtain the sampling distribution for the mean and variance of a normal distribution when
N = 2 scores in each sample. What the applet is going to do is create 10,000 random samples, each
with two scores, drawn from a normally distributed population. Then it will compute the mean and the
variance for each sample. Then it will, for each of resulting two sampling distributions, give you some
basic statistics and a plot of the sampling distribution.
The parent population here is
normal with mean 16,
variance 25.
Set the sample size to 2, ask
for distributions of the sample
means and the sample
variances [Var(U)] and check
“Fit normal.” Then click on
“10,000.”
Look at the distribution of
means. The shape of the
distribution closely
approximates that of the
superimposed normal
distribution and, as expected,
the mean of the means is
nearly equal to the
population mean (it would be
exactly equal if we ran many
more replications). The
sample mean is an unbiased
estimator.
M
You should expect the standard deviation of the distribution of sample means to be

5


 3.536. Your simulated standard error should be close to this.
N
2
Now look at the distribution of the sample variances. The mean of the 10,000 sample
variances will be nearly equal to the population variance -- the sample variance is an unbiased
estimator. Notice, however, that it is very skewed. The median is much lower than the mean; over
half of the sample variances are less than the population variance.
Now change the statistic for the lower plot from Var(U) to Variance. Var(U) is the sample
variance, with (N-1) in the denominator. For “Variance” the denominator is N. Click on 10,000 again.
You will observe that the mean of the sampling distribution is now very far from the population
variance. As I noted early in the semester, reducing N by one is necessary if you want to have an
unbiased estimator of the population variance.
Switch back to the unbiased estimator and change the sample size to 25. Click 10,000 again.
Notice that the standard
errors, for both the mean and
the sample variance, have
dropped with the increase in
sample size – both are
consistent estimators.
Also notice that the shape of
the distribution of sample
variances is approaching that
of the normal distribution.
I encourage you to use this
applet to play with other
statistics and other
characteristics – for example,
investigate the effect of
shape of parent population
and sample size. On the
next page I have simulated
the sampling distributions of
the mean and of the median.
You already know that the
median is more resistant to
the influence of outlier than is
the mean. You will find,
however, that we shall
employ the mean much more
often than the median. Why?
Look at the simulation results
to the right. The standard
error of the mean is
5/SQRT(25) = 1 and the
standard error of the median
is 1.27. That is, the mean is
a more efficient estimator
than is the median. Put
another way, there is more
error in the estimation of the
median than there is in the
estimation of the mean.

This applet is also available at
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
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