Standard error and bias

When the sample mean is used to estimate a population mean, µ, and the population standard deviation is σ, the error distribution is approximately

error  ~  normal (0,   )

Since the error distribution is centred on zero, the estimator is called unbiased.

bias  =  μerror  =  0

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

standard error  =  σerror  = 

Important property

The error distribution does not depend on the value of the parameter that we are estimating, µ.

As a result, we can find the error distribution in practical problems provided the population standard deviation, σ, is known.

Active ingredient in medicine

Pharmaceutical companies routinely test their products to ensure that the concentration of active ingredient, µ, is within tight limits. However the chemical analysis is not precise and repeated measurements of the same specimen differ slightly.

One type of analysis gives estimated concentrations of the active ingredient that are normally distributed with standard deviation σ = 0.0068 grams per litre. When a product is tested once, the recorded concentration is therefore

X  ~  normal (μ , σ = 0.0068)

A product is tested several times, giving a sample mean concentration .

The sample mean is our best estimate of the population mean, but how accurate is the estimate?

From a sample of size n, the estimation error has distribution,

error  ~  normal (0,  σ = )

This is illustrated in the following diagram.

Example

Say a particular product was tested 16 times, giving a sample mean concentration of 0.0724 grams per litre. Drag the slider above to show the error distribution for samples of size n = 16.

Our estimate of 0.0724 grams per litre is unlikely to be more than 0.004 from the true concentration for this product.