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.

Soluble sugar in plants

A laboratory procedure for assessing soluble sugar in plants is such that in the range 100 to 200 milligrams (mg) of glucose per gram of dry weight, repeat measurements will follow a normal distribution with mean µ equal to the true glucose level and standard deviation σ = 3.0 mg/g dry weight.

When a plant is tested once, the recorded glucose content is therefore

X  ~  normal (μ , σ = 3)

A plant is tested several times, giving a sample mean glucose level .

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 plant was tested 16 times, giving a sample mean of 137 mg of glucose per gram dry weight. Drag the slider above to show the error distribution for samples of size n = 16.

Our estimate of 137 mg/g dry wt is unlikely to be more than 1.5 from the true glucose content for this plant.