"It has been proved beyond a shadowof a doubt that smoking is one of the leading causes ofstatistics."

Fletcher Knebel

9.1 SAMPLING DISTRIBUTIONS (Pages 456 - 469)

OVERVIEW: One examines samples inorder to come to reasonable conclusions about the population fromwhich the sample is chosen. One must be statistically literate inorder to gleen meaningful information from a sample. This involves anawareness of what the sample results tell us, along with what theydon't tell us. A statistic calculated from a sample may suffer frombias or high variability, and hence not represent a good estimate ofa population parameter.

Parameter: An index that isrelated to a population.

Statistic: An index that isrelated to a sample.

Sampling distribution of a statistic: The distribution of values of a statistic taken from allpossible samples of a specific size.

A statistic is unbiasedif the mean of the sampling distribution isequal to the true value of the parameter being estimated.

A reminder that

s2 =[sum(xi-mean)2]/N ... the variance formula for a population.

s2 = [sum(xi-mean)2]/(N-1)... the varianceformula for a sample.

The following example demonstrates some of thestatistical concepts developed in this section.

Consider the three element population P={1,2,3}.

The mean of P is m = 2

The standard deviation of P is s = 0.81649658

The variance of P is s2 =0.66666667

These values are parameters, since they arederived from a population.

Now, consider all possible samples of size 2, withreplacement. There would be 32 = 9 such samples.

 Sample Sample Mean Sample Var. = s2 Sample St.Dev.=s 1,1 1 0 0 1,2 1.5 .5 .707107 1,3 2 2 1.4142 2,1 1.5 .5 .707107 2,2 2 0 0 2,3 2.5 .5 .707107 3,1 2 2 1.4142 3,2 2.5 .5 .707107 3,3 3 0 0 MEANS 2.0 0.66666667 0.628539

The table shows that...

-the mean ofthe distribution of sample means is themean ( m)of the population. This illustrates that a sample mean is anunbiased estimator of the population mean. (The distribution of sample means"centers" around the mean of the population.)

-the mean of thedistribution of sample variances(s2) isequal to the variance ( s2) of thepopulation. This illustrates that a sample variance(s2) is anunbiased estimator of the population variance. (The distribution of samplevariances "centers" around the variance of the population.)

Note: A sample standard deviation isnot an unbiasedestimator of the population standard deviation. In this example, themean of the sample standard deviations (s) is 0.628539, and thestandard deviation of the population is s = 8.81649658. (Thedistribution of sample standard deviations does "not center" aroundthe standard deviation of the population.)