Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS

or all of the quantitiesaand they may be unknown. When this occurs, one must

substitute estimated values of any unknown quantities into the expression forσaˆ

in order to obtain an estimated standard errorσˆˆa. One then quotes the result as

a=aˆ±σˆˆa.

Ten independent sample valuesxi,i=1, 2 ,..., 10 , are drawn at random from a Gaussian

distribution with standard deviationσ=1. The sample values are as follows (to two decimal

places):

2 .22 2.56 1.07 0.24 0.18 0.95 0. 73 − 0 .79 2.09 1. 81

Estimate the population meanμ, quoting the standard error on your result.

We have shown in the final worked example of subsection 31.3.1 that, in this case, ̄xis
a consistent, unbiased, minimum-variance estimator ofμand has varianceV[ ̄x]=σ^2 /N.
Thus, our estimate of the population mean with its associated standard error is

ˆμ=x ̄±

σ

√

N

=1. 11 ± 0. 32.

If the true value ofσhad not been known, we would have needed to use an estimated
valueσˆin the expression for the standard error. Useful basic estimators ofσare discussed
in subsection 31.4.2.

It should be noted that the above approach is most meaningful for unbiased

estimators. In this case,E[aˆ]=aand soσˆadescribes the spread ofaˆ-values about

the true valuea. For a biased estimator, however, the spread about the true value

ais given by theroot mean square erroraˆ, which is defined by

^2 aˆ=E[(aˆ−a)^2 ]

=E[(aˆ−E[ˆa])^2 ]+(E[ˆa]−a)^2

=V[aˆ]+b(a)^2.

We see that^2 aˆis the sum of the variance ofaˆand the square of the bias and so

can be interpreted as the sum of squares of statistical and systematic errors. For

a biased estimator, it is often more appropriate to quote the result as

a=aˆ±aˆ.

As above, it may be necessary to use estimated valuesaˆin the expression for the

root mean square error and thus to quote only an estimateˆaˆof the error.

31.3.4 Confidence limits on estimators

An alternative (and often equivalent) way of quoting a statistical error is with a

confidence interval. Let us assume that,otherthan the quantity of interesta,the

quantitiesahave known fixed values. Thus we denote the sampling distribution