Statistical Methods for Psychology

derived mathematically, but it is easier to understand what they represent if we consider

how they could, in theory, be derived empirically with a simple sampling experiment.

We will take as an illustration the sampling distribution of the differences between

means,because it relates directly to our example of waiting times in parking lots. The sam-

pling distribution of differences between means is the distribution of differences between

means of an infinite number of random samples drawn under certain specified conditions

(e.g., under the condition that the true means of our populations are equal). Suppose we

have two populations with known means and standard deviations (Here we will suppose

that the two population means are 35 and the population standard deviation is 15, though

what the values are is not critical to the logic of our argument. In the general case we rarely

know the population standard deviation, but for our example suppose that we do.) Further

suppose that we draw a very large number (theoretically an infinite number) of pairs of ran-

dom samples from these populations, each sample consisting of 100 scores. For each sam-

ple we will calculate its sample mean and then the difference between the two means in

that draw. When we finish drawing all the pairs of samples, we will plot the distribution of

these differences. Such a distribution would be a sampling distribution of the difference be-

tween means. I wrote a 9 line program in R to do the sampling I have described, drawing

10,000 pairs of samples of n 5 100 from a population with a mean of 35 and a standard

deviation of 15 and computing the difference between means for each pair. A histogram of

this distribution is shown on the left of Figure 4.1 with a Q-Q plot on the right. I don’t think

that there is much doubt that this distribution is normally distributed. The center of this dis-

tribution is at 0.0, because we expect that, on average, differences between sample means

will be 0.0. (The individual means themselves will be roughly 35.) We can see from this

figure that differences between sample means of approximately 2 3 to 1 3, for example,

are quite likely to occur when we sample from identical populations. We also can see that

it is extremely unlikely that we would draw samples from these populations that differ by

10 or more. The fact that we know the kinds of values to expect for the difference of means

of samples drawn from these populations is going to allow us to turn the question around

and ask whether an obtained sample mean difference can be taken as evidence in favor of

the hypothesis that we actually are sampling from identical populations—or populations

with the same mean.

Section 4.2 Sampling Distributions 89

Q-Q plot for normal sample

10,000 samples representing

Ruback and Juieng study

Obtained quantiles

–2 –1 021

Expected quantiles

–2

–1

0

1

2

–6 –4 –2^024

Difference in mean waiting times

6

800

600

400

200

Frequency

0

Obtained

mean

Figure 4.1 Distribution of difference between means, each based on 25 observations

sampling
distribution of
the differences
between means