# Estimating the Population Mean ( ) and Variance

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Estimating the Population Mean (  ) and Variance ( 2)1

George H Olson, Ph. D. Leadership and Educational Studies
Appalachian State University

Unbiased estimates of population parameters
A statistic, w, computed on a sample, is an unbiased estimate of a population parameter, θ, if its expected value [ℰ (w)] is the parameter, θ.
Unbiased estimate of the population mean

It is easy to show, for example, that the sample mean, X , provides an unbiased estimate of the population mean, μ.

 X    X1  X 2  XN3   X N  (1)
  X1   X2    X3    XN .
N

However, any ℰ ( X i ) is, by definition, µ, for all observations taken from the same population.
Therefore,

X  N (X)  .

(2)

N

Unbiased estimate of the population variance
In contrast, it can be shown that the sample variance, s2, is a biased estimate of the population variance, σ2, i.e., that ℰ (S2) ≠ σ2:

 

N

X i2

 

 s2   i

 X2

N

(3)

 

N

X i2

 

    i  

N

X2 .

Consider, first, the first term to the right of the equal sign in (3), above.

1 The material presented here is derived from Hays (1973, 2nd Ed, pp. 272-274

    

N

X i2

 

N

X i2

 i  i

.

(4)

N

N

By definition, the population variance is,

2  X2  2;

(5)

so that, for any observation, i,

 Xi2    2  2 .

(6)

Substituting (6) into (4) yields,

 

N

X i2

 

N

X

2 i

   i   i

N

N

(7)

N

2  2

i N

  N  2  2
 N

 2  2.

Now, consider the second term to the right of the equal sign in (3), sampling distribution of means is:

 X2  

 X 2   [  X ]2 , X2  2

 X 2  . The variance of the
(8)

from which,

X

2

2 X

2 .

(9)

Substituting the expressions, (7) and (9) into (3) yields:

s2  

 

N

X i2

 

 i 

N

X2

      2  2   2  2

(10)

X

  2   X2 .

In words, the expected value of the sample variance is the difference between the population variance,  2 , and the variance of the distribution of sample means,  X2 . Since the variance of the distribution of sample means typically is not zero, the sample variance under-estimates the population variance. In other words, the sample variance is a biased estimator of the population variance.

It has already been demonstrated, in (2), that the sample mean, X , is an unbiased estimate of the
population mean, µ. Now we need an unbiased estimate ( s 2 ) {note the tilde to imply estimate} of the population variance σ2. In (10), it was shown that

s2



2

 

2 X

,

which, after substituting (11), yields

(s2)   2   2 N

 N2 2

(11)

N

N

 

N

 1

2.

N

Equation (13) shows that the average of the sample variances [ (s2 ) ] is too small by a

factor of N . N 1
Hence, an unbiased estimate of  2 is given by

sˆ2  N s2. (N 1)

It is easy to show that sˆ2 is an unbiased estimate of  2 :

 sˆ2

 

N

 

(s2)

 N 1

 

N

 

N

 1  

2

 N 1 N 

  2.

Unbiased estimate of the standard error of the mean, X .

The unbiased estimate of  X is given by sˆX , where

sˆ  sˆ2

X

N

1 

N

sˆ2

 N  (N 1)

(12)

 s2 N

 sˆ . N

In words, the unbiased estimate of the standard error of the mean is the unbiased estimate of the population standard deviation divided by the square root of the sample size.
EstimatePopulationPopulation VarianceSampleVariance