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.