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.
 XN 
N
E  X1   E  X 2   E  X 3  

N
E X  E
 X1  X 2  X 3 
(1)
E XN 
.
However, any ℰ ( X i ) is, by definition, µ, for all observations taken from the same population.
Therefore,
E X 
NE ( X )
.
N
(2)
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 2

  Xi

E  s2   E  i
 X2
 N





 N 2
  Xi 
  E  X 2 .
E  i
 N 




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, 2 nd Ed, pp. 272-274
(3)
 N 2 N
2
  X i  E  X i 
 i
E i
.
N
N






(4)
By definition, the population variance is,
 2  E  X 2   2 ;
(5)
so that, for any observation, i,
E  X i2    2   2 .
(6)
Substituting (6) into (4) yields,
 N 2 N
2
  X i  E  X i 
 i
E i
N
 N 




 
N

2

2

(7)
i
N
N    2 
2

N
    2.
2
Now, consider the second term to the right of the equal sign in (3), E  X 2  . The variance of the
sampling distribution of means is:
 X2  E  X 2   [E  X ]2
 E  X 2   2
,
(8)
from which,
E  X 2    X2   2 .
Substituting the expressions, (7) and (9) into (3) yields:
(9)
 N 2
  Xi 
2
 E X 2
E s   E  i
 N 




  2   2    X2   2 
(10)
  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
E  s 2    2   X2 ,
which, after substituting (11), yields
E (s 2 )   2 
2
N
N 2 2
 
N
N
 N 1 2

 .
 N 

(11)
Equation (13) shows that the average of the sample variances [ E ( s 2 ) ] is too small by a
factor of
N
.
N 1
Hence, an unbiased estimate of  2 is given by
sˆ 2 
N
s2 .
( N  1)
It is easy to show that ŝ 2 is an unbiased estimate of  2 :
 N 
2)
E sˆ 2  
 E (s
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 sˆ X , where
sˆX 
sˆ 2
N
1 N
  
sˆ 2
 N  ( N  1)


(12)
2
s
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.