Statistical Methods for Psychology

given experiment) rather than individual observations or events. We have already discussed

sampling distributions in Chapter 4, and what we said there applies directly to what we will

consider in this chapter.

The Mean and Variance of a Binomial Distribution

In Chapter 2, we saw that it is possible to describe a distribution in many ways—we can

discuss its mean, its standard deviation, its skewness, and so on. From Figure 5.4 we can

see that the distribution for the outcomes for our judge is symmetric. This will always be

the case for p 5 q 5 .50, but not for other values of pand q. Furthermore, the mean and

standard deviation of any binomial distribution are easily calculated. They are always:

For example, Figure 5.4 shows the binomial distribution when N 5 10 and p 5 .50. The

mean of this distribution is 10(.5) 5 5 and the standard deviation is

We will see shortly that being able to specify the mean and standard deviation of any

binomial distribution is exceptionally useful when it comes to testing hypotheses. First,

however, it is necessary to point out two more considerations.

In the example of perception without awareness, we assumed that our judge was choos-

ing at random (p 5 q 5 .50). Had we slowed down the stimulus so as to increase the per-

son’s accuracy of response on any one trial—the arithmetic would have been the same but

the results would have been different. For purposes of illustration, three distributions

obtained with different values of pare plotted in Figure 5.5.

1 2.5=1.58.

1 10(.5)(.5) =

Standard deviation= 2 Npq

Variance=Npq

Mean=Np

130 Chapter 5 Basic Concepts of Probability

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Probability

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

Number of successes

p = 0.60 p= 0.30 p = 0.05

Figure 5.5 Binomial distributions for N 5 10 and p 5 .60, .30, and .05