Standard Error of the Mean

When trying to work out the mean of a sample, we may have to answer one of two questions:

If there is no control over the size of the sample, what is the confidence of the mean.

If the size of the sample can be controlled, how big a sample should be taken to ensure that the confidence interval is within the required limits.

The solution is based on the standard error of the mean.  The distribution of sample means is normally distributed, regardless of of the distribution of the population from which the sample was drawn.  The standard error of the mean, is the standard deviation of the sample mean.  The formula is:

It follows from this equation, that the larger the sample size, the smaller the likely error.  This is one of those cases where statistics tells you something that is intuitively obvious.

In practice we don't always know the standard deviation of the population (or a value calculated from a very large sample), so we may have to estimate the value from the sample.

The confidence limits of the sample mean are:

Example - Confidence Interval for the Mean.

The standard deviation of the weight of individual chilies from this years crop was 2.0 grams.  The average weight of a sample of 16 chilies was 10 grams.  Thus the standard error of the mean is:

StErr = 2/√16 = 0.5

Therefore the 95% confidence interval for the mean is:

Confidence interval = 10 ± 1.96 * 0.5 = 10 ± 0.98 = 9.02 to 10.98

Sample Size

The formula for the standard error can be re-arranged to estimate the required sample size:

Thus if we know the confidence level and the interval size, we can work out the standard error to plug into the equation:

Example - Sample Size

Back to our crop of chilies, say, we need to be 95% certain that our estimate of the mean has an error of plus or minus 0.50 grams, how big should the sample be.

Step 1 - Calculate the standard error

StErr = interval/1.96 = 0.5/1.96 = 0.26

Step 2 - Work out the sample size

n = (StDev/StErr)² = (2.00/0.26)² = 7.69² = 59.1

In practice, we would take the next integer value which is greater then 59.1, i.e. 60 as the required sample size.

Theory & Practice

An "experiment" illustrates the theory.  A computer program was used to select 1,000 samples of 100 numbers from two different populations.  The first was a normal distribution and the second a highly left skewed lognormal.  Whilst both had the same mean and standard deviation, there were large differences in the range, mode and median.

The parameters were contrived to demonstrate the distribution of the sample means would be the same.

1,000 Samples from a Normal Distribution (size=100)

1,000 Samples from a LogNormal Distribution (size=100)

For both distributions, the standard error of the mean is:

The histograms of the distribution of the sample means are similar and have the familiar bell shape of the normal distribution and the mean and standard deviation are also similar:

	Mean	StDev
Normal	9.04	0.90
LogNormal	9.04	0.89

Application

One application of the standard error of the mean is in Monte-Carlo process where random numbers are thrown at an algorithm, the results of which are then sumarised to provide a range of outcomes.  One way of determining the number of simulations to perform is to monitor the standard error of one or more output parameters, If this falls within acceptable limits the process is terminated.

Page Updated: 11-Feb-2008