Standard error and bias

When the sample mean is used to estimate a population mean, µ, the estimator is unbiased.

error  =  μerror  =  0

The estimator's standard error is the standard deviation of the error distribution,

standard error  =  σerror  = 

Note that the error distribution does not depend on the value of µ. We can therefore evaluate the standard error provided the value of σ is known.

Example

We know that a particular type of measurement, X, has a normal distribution with known standard deviation σ = 0.0068 but unknown mean µ. A random sample of n = 16 values has sample mean 0.0724 and this is used to estimate µ . We can then find the distribution of errors that will arise using:

error  ~  normal (0,  σ = )

For sample size n = 16, this error distribution is: