Standard Error
The standard error of a statistic is an estimate of the standard deviation of a
statistic. The standard deviation of a statistic describes how a statistic
varies from sample to sample.
For example, suppose we will take a simple random sample of size 2 from the
following population.
Population: 1, 3, 4, 5, 7
There are many possible samples, so there are many possible values of the sample
mean Y-bar. The table below lists all the possibilities.
Sample Y-bar
1, 3 2.0
1, 4 2.5
1, 5 3.0
1, 7 4.0
3, 4 3.5
3, 5 4.0
3, 7 5.0
4, 5 4.5
4, 7 5.5
5, 7 6.0
The standard deviation of Y-bar is the standard deviation of the numbers in the
Y-bar column.* The standard deviation tells how Y-bar varies from sample to
sample. The nice thing is that we can estimate the standard deviation of Y-bar
using only a single sample (which is all we ever get to see in a real problem).
For example, suppose we draw the sample
Sample: 3, 7
Y-bar = (3+7)/2 = 5
s = sqrt{ [ (3-5)^2 + (7-5)^2 ] / (2-1) } = sqrt(8)
The standard error of Y-bar is estimated to be
s/sqrt(n)=sqrt(8)/sqrt(2)=2
This is an estimate of the standard deviation of the numbers in the Y-bar column
in the table above.
*FOOTNOTE The standard deviation of the sample mean is approximately
sigma/sqrt(n) if the number of items in the population is much larger than the
number of items sampled. In a simple example like this one where the population
is very small, the standard deviation of Y-bar won't be sigma/sqrt(n). This is
nothing to worry about. The purpose of the example is to help you better
understand the standard deviation of a statistic and the standard error of a
statistic.