Sampling Distribution of Difference Between Means

Prerequisites
Sampling Distributions, Sampling Distribution of the Mean, Variance Sum Law I

Learning Objectives

  1. State the mean and variance of the sampling distribution of the difference between means
  2. Compute the standard error of the difference between means
  3. Compute the probability of a difference between means being above a specified value

The sampling distribution of the difference between means can be thought of as the distribution that would result if we repeated the following three steps over and over again: (1) sample n1 scores from Population 1and n2 scores from Population 2, (2) compute the means of the two samples (M1 and M2), (3) compute the difference between means M1 - M2. The distribution of the differences between means is the sampling distribution of the difference between means.

As you might expect, the mean of the sampling distribution of the mean is:

which says that the mean of the distribution of differences between sample means is equal to the difference between population means. For example, say that mean test score of all 12-year olds in a population is 34 and the mean of 10-year olds is 25. If numerous samples were taken from each age group and the mean difference computed each time, the mean of these numerous differences between sample means would be 34 - 25 = 9.

From the variance sum law, we know that:

which says that the variance of the sampling distribution of the difference between means is equal to the variance of the sampling distribution of the mean for Population 1 plus the variance of the sampling distribution of the mean for Population 2. Recall the formula for the variance of the sampling distribution of the mean:

Since we have two populations and two samples sizes, we need to distinguish between the two variances and sample sizes. We do this using the subscripts 1 and 2. Using this convention we can write the formula for the variance of the sampling distribution of the difference between means as:

Since the standard error of a sampling distribution is the standard deviation of the sampling distribution, the standard error of the difference between means is:

Just to review the notation, the symbol on the left contains a sigma (σ) which means it is a standard deviation. The subscripts M1 - M2 indicate that it is the standard deviation of the sampling distribution of M1 - M2.

As shown below, the formula for the standard error of the difference between means is much simpler if the sample sizes and the population variances are equal. Since the variances and samples sizes are the same, there is no need to use the subscripts 1 and 2 to differentiate these terms.