Sampling Distribution of Difference Between
Means
Prerequisites
Sampling
Distributions, Sampling
Distribution of the Mean, Variance
Sum Law I
Learning Objectives
 State the mean and variance of the sampling distribution of the difference
between means
 Compute the standard error of the difference between means
 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 12year olds in a population is 34 and the mean of 10year
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.
