Statistical Methods for Psychology

the probabilities for each outcome between 0 and 10 correct out of 10, we would find the

results shown in Table 5.4. Observe from this table that the sum of those probabilities is 1,

reflecting the fact that all possible outcomes have been considered.

Now that we have calculated the probabilities of the individual outcomes, we can plot

the distribution of the results, as has been done in Figure 5.4. Although this distribution re-

sembles many of the distributions we have seen, it differs from them in two important

ways. First, notice that the ordinate has been labeled “probability” instead of “frequency.”

This is because Figure 5.4 is not a frequency distribution at all, but rather is a probability

distribution. This distinction is important. With frequency, or relative frequency, distribu-

tions, we were plotting the obtained outcomes of some experiment—that is, we were plot-

ting real data. Here we are not plotting real data; instead, we are plotting the probability

that some event or another will occur.

To reiterate a point made earlier, the fact that the ordinate (Y-axis) represents probabili-

ties instead of densities (as in the normal distribution) reflects the fact that the binomial

distribution deals with discrete rather than continuous outcomes. With a continuous distri-

bution such as the normal distribution, the probability of any specified individual outcome

is near 0. (The probability that you weigh 158.214567 pounds is vanishingly small.) With a

discrete distribution, however, the data fall into one or another of relatively few categories,

and probabilities for individual events can be obtained easily. In other words, with discrete

distributions we deal with the probability of individual events, whereas with continuous

distributions we deal with the probability of intervals of events.

The second way this distribution differs from many others we have discussed is that al-

though it is a sampling distribution, it is obtained mathematically rather than empirically.

The values on the abscissa represent statistics (the number of successes as obtained in a

Section 5.8 The Binomial Distribution 129

Table 5.4 Binomial distribution for p 5 .50, N 510

Number Correct Probability

0 .001

1 .010

2 .044

3 .117

4 .205

5 .246

6 .205

7 .117

8 .044

9 .010

10 .001

1.000

012345678910

Number correct

Probability

0.25

0.20

0.15

0.10

0.05

0

Figure 5.4 Binomial distribution when N 5 10 and p 5 .50