Sanderson M. Smith
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Consider the number set P = {2,4,6}.
If P is a population, then
Mean = m = (2+4+6)/3 = 4.
Variance = s^{2} = [(24)^{2} + (44)^{2} + (64)^{2}]/3 = 8/3 = 2.666667.
Standard deviation = sqrt(s^{2}) = sqrt(8/3) = s = 1.632993.
NOTE: The formula for s^{2} involves dividing by the population size n. In this case, n = 3.
If P is a sample, then
Sample mean = x(bar) = (2+4+6)/2 = 4.
Sample variance = s^{2} = [(24)^{2} + (44)^{2} + (64)^{2}]/2 = 8/2 = 4.
Sample standard deviation = sqrt(s^{2}) = sqrt(4) = s = 2.
NOTE: The formula for s^{2} involves dividing by n1. In the case, n=3. Hence n1 = 2.
Now, let's consider P to be a population.
The table below shows all possible samples of size 2 chosen from P, with replacement. There would be 3x3 = 9 samples.
Sample 
x(bar) for sample 
s^{2} for sample 
s for sample 
s^{2} for sample 
s for sample 




























































Column Means 





To summarize, we have listed all samples of size 2 (with replacement) from a population P of size 3. We have calculated statistics for each sample of size 2. Here is an important definition:
A statistic used to estimate a population parameter is unbiased if the mean of the sampling distribution of the statistic is equal to the true value of the parameter being estimated.
Note that
The mean of the sample means (4) is equal to m, the mean of the population P. This illustrates that a sample mean x(bar) is an unbiased statistic. It is sometimes stated that x(bar) is an unbiased estimator for the population parameter m .
The mean of the sample values of s^{2} (2.666667) is equal to s^{2} , the variance of the population P. This illustrates that the sample variance s^{2} is an unbiased statistic. It is sometimes stated that s^{2} is an unbiased estimator for the population variance s^{2.}
Note carefully that the sample statistic s is not an unbiased statistic. That is, the mean of the s column in the table (1.257079) is not equal to the population parameter s = 1.632993.
Also, if you use the s^{2} formula for samples, the resulting statistics are not unbiased estimates for a population parameter. Note that the means for the last two columns in the table are not equal to population parameters.
In summary, the sample statistics x(bar) and s^{2} are unbiased estimators for the population mean m and population variance s^{2}, respectively.
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