Standard Error
Inferential statistics are based on the concept of making decisions using distributions
of sample means (AKA sampling distributions). The reason to use samples is that
populations are usually too large or impossible to test.
If one sets up a distribution of means, you can judge the relative position of your sample
mean compared to the population mean. With this information, you can make a
judgment called a hypothesis test.
In any case - there are different shapes of sampling distributions.
Some are bell shaped like the t-distribution. Some have asymmetrical shapes like the
Chi-square and F-distributions (both related).
You generate sampling distribution by:
 Defining a population
 Generating all possible samples of a given size from the population
 Plotting that distribution.
We will use an artificial population of five numbers, which are 1, 2, 3, 4, 5. If you want,
assume that it is after World War III and only 5 people survive, they are the population.
We test them on some variable. One person gets 1, another gets 2, etc.
Note that the mean () for this population is 3 and
its standard deviation is 1.414.
I decide to take a sample size of 2. I pick a person at random. Then, I pick again. It is
possible that I pick the same person twice. These would be all my possible samples. I
have also calculated the mean of each of the samples. There are 25 possible samples.
Subject 1
1.00
1.00
1.00
1.00
1.00
2.00
2.00
2.00
Subject 2
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
Average of The Sample*
1.00
1.50
2.00
2.50
3.00
1.50
*Average = (2 + 1)/2 = 1.50
2.00
2.50
2.00
2.00
3.00
3.00
3.00
3.00
3.00
4.00
4.00
4.00
4.00
4.00
5.00
5.00
5.00
5.00
5.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
3.00
3.50
2.00
2.50
3.00
3.50
4.00
2.50
3.00
3.50
4.00
4.50
3.00
3.50
4.00
4.50
5.00
To have my distribution of means, I prepare a frequency distribution of the sample
means (my last column of figurers).
Sample Average
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Frequency
1
2
3
4
5
4
3
2
1
Percent
4.0
8.0 -> Look for the two 1.50 values above
12.0
16.0
20.
16.0
12.0
8.0
4.0
Total Number of Samples = 25
Now, I graph this distribution and calculate its mean and standard deviation.
- This is a distribution of means.
Its mean has to be the same as that of the population mean. Its standard deviation is the
standard error of the mean. Look at the Graphic below.
Thus, we can see
that most samples
are close to 3.0 but
there are some
extreme sample
means with values of
1.0, 1.5 and 4.5 and
5.0.
Now I will redo the
exercise with a
sample size of 3.
Here is the list of all
possible samples of
size 3, with averages
for each sample:
Subject 1
Subject 2
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2.00
1.00
2.00
1.00
2.00
1.00
2.00
1.00
2.00
1.00
3.00
1.00
3.00
1.00
3.00
1.00
3.00
1.00
3.00
Etc., etc., etc.
1.00
5.00
1.00
5.00
1.00
5.00
1.00
5.00
2.00
1.00
2.00
1.00
2.00
1.00
2.00
1.00
2.00
1.00
2.00
2.00
2.00
2.00
2.00
2.00
Etc., etc., etc.
2.00
4.00
2.00
4.00
2.00
4.00
2.00
4.00
Subject 3
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
Average of Sample
1.00
1.33
1.67
2.00
2.33
1.33
1.67
2.00
2.33
2.67
1.67
2.00
2.33
2.67
3.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
2.67
3.00
3.33
3.67
1.33
1.67
2.00
2.33
2.67
1.67
2.00
2.33
1.00
2.00
3.00
4.00
2.33
2.67
3.00
3.33
2.00
4.00
2.00
5.00
2.00
5.00
2.00
5.00
2.00
5.00
2.00
5.00
3.00
1.00
3.00
1.00
3.00
1.00
3.00
1.00
3.00
1.00
Etc., etc., etc.
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
Sample Average
1.00
1.33
1.67
2.00
2.33
2.67
3.00
3.33
3.67
4.00
4.33
4.67
5.00
5.00
1.00
2.00
3.00
4.00
5.00
1.00
2.00
3.00
4.00
5.00
3.67
2.67
3.00
3.33
3.67
4.00
1.67
2.00
2.33
2.67
3.00
1.00
2.00
3.00
4.00
5.00
3.67
4.00
4.33
4.67
5.00
Frequency
1
3
6
10
15
18
19
18
15
10
6
3
1
Percent
.8
2.4
4.8
8.0
12.0
14.4
15.2
14.4
12.0
8.0
4.8
2.4
.8
Now, I present a graphic which compares the sampling distribution with a
sample size of two versus a sample size of three. The frequencies of the means are
presented on a percentage basis for easy comparison.
1. Fewer Extremes: With the larger sample size (3) - note that there are fewer
extreme sample means. Look at the number of samples means with a mean of 5.
This is an extreme and not very representative mean. The percentage is dramatically
less for the N=3 sample.
Thus, with larger samples - you don't get wacky means that much.
2. Tighter distributions - note that the standard deviation of all the sample means (the
standard error) is smaller than with a sample size of 2. It's mean is again 3 but the
standard deviation of this distribution of means is equal to .82.
Sampling distributions of the mean and those of some other statistics have a
particular shape. They are bell shaped, like the normal curve, but less peaked and
with fatter tails.
This particular shape is called leptokurtic (from leptokurtosis).
The fatness of the tails is controlled by a parameter called the degrees of freedom (df).
DF are related to sample size. In our example, df = Nsample -.1. So for example with 3
subjects in the sample, df =2.
The graphics above are actual frequency distributions. However, the t-distributions are
theoretical mathematical functions. Here are some examples comparing the t-
distributions with df = 3 or 6. Note the fatter tails and cut off scores for 5% total
extremes (2.5% in each tail). t-distributions have cut offs like the z-score in Workshop
2.
Let's look at the tails of the distributions. We've marked the cut offs for the 5% two
tailed level.
.
This reflects
our example
above where
the smaller
sample size gave more extreme sample means. Thus, with big samples it's hard to get
weird means.
Depending on your situation, you don't have to actually construct a distribution. The
value of the standard error can be calculated.
equals the standard deviation calculated from your sample.
The standard error formula can vary for different sample statistics. One can determine
the standard error for a proportion, different between means, a correlation coefficient,
slope of a regression line, intercept and other items. Your texts can supply these.
The important idea is that you have a sample statistic and want to build a frequency
distribution of all the possible samples. Then, you want to describe the variability of
these samples. This is the use of the standard error.
Here are some important points about samples:
Samples
Representative?
You want your
sample to mirror
the important
characteristics of
the population.
What Size Sample
Do I Need?
How Many Samples
Do I Need?
The larger the sample,
the more likely it will
be representative. For
most situations, there
are methods to
calculate the
appropriate sample
size.
A Sampling Distribution
is based on the idea of all
possible samples.
However, you only need
to calculate one sample to
get the statistics you need
to use the sampling
distribution.