DRAWING A CONCLUSION

Now that you have seen how to use the standard error of a difference in means, how do you compute it? You take the square root of the variance of the difference. How do you get the variance of the difference? When you have two means from independent samples, the variance of their difference equals the sum of their variances. This neat little fact would take too long to prove here, but you can see how it is used. The example above had two means from independent samples of size 20, taken from a population whose standard deviation was 15. You calculate the standard error of the difference by following these steps:

• As explained earlier, the standard error of each mean equals 15 (the standard deviation) divided by the square root of 20 (the sample size).

• The variance of each mean is just the square of that fraction: 15 squared divided by 20. That is 225 divided by 20, or 11.25.

• The variance of the difference between the means is the sum of the variances of each mean. That is 11.25 plus 11.25, or 22.5.

• The standard error of the difference between the means is therefore the square root of 22.5, or 4.74.