# Estimating the Population Mean ( ) and Variance

## Transcript Of Estimating the Population Mean ( ) and Variance

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

N2 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.

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

N2 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.