THE STANDARD ERROR OF THE MEAN

If the mean of the distribution of sample means is the population mean, what is the standard deviation of the distribution of sample means? Is it also just the standard deviation of the population? No. As previously mentioned, the standard deviation of the means depends on two things:

• How large a sample you take. Larger samples mean a smaller standard deviation for the sample means.

• How much variability exists in the population. Less variability in the samples also means a smaller standard deviation for the sample means.

To calculate the exact standard deviation of the distribution of sample means, you must know:

• The standard deviation in the population.

• The number of cases in the sample.

All you have to do is divide the standard deviation by the square root of the sample size . The result, the standard deviation of the distribution of sample means, is called the standard error of the mean. Although it has an impressive name , it is still just a standard deviation ” the standard deviation of the sample means. Think about the formula for computing the standard error of the mean: take the standard deviation of the variable and divide by the square root of the sample size. Suppose the standard deviation of number of books owned is 50, and the sample size is four cases. Then the standard error is 50 divided by the square root of 4, to yield 25. If the sample size is increased to 9, the standard error decreases to 50 divided by the square root of 9, or 16.7. If the sample size is increased to 100, the standard error is only 5. The larger the sample size, the less variability there is in the sample means.