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:
A statistic is unbiasedif the mean of the sampling distribution isequal to the true value of the parameter being estimated.
A reminder that
s
2 =[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 3
Sample | Sample Mean | Sample Var. = s2 | Sample St.Dev.=s |
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MEANS | | | |
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.)