Statistical Methods for Psychology

presents the distribution of responses for several of these aspects. The possible values of X

(the rating) are presented on the abscissa (X-axis), and the relative frequency (or probabil-

ity) of people choosing that response is plotted on the ordinate (Y-axis). From the figure

you can see that the distributions of responses to questions concerning health, friends, and

savings are quite different. The probability that a person chosen at random will consider

his or her health to be extremely important is .70, whereas the probability that the same

person will consider a large bank account to be extremely important is only .16. (So much

for the stereotypic American Dream.) Campbell et al. collected their data in the mid-1970s.

Would you expect to find similar results today? How may they differ?

5.5 Probability Distributions for Continuous Variables

When we move from discrete to continuous probability distributions, things become more

complicated. We dealt with a continuous distribution when we considered the normal dis-

tribution in Chapter 3. You may recall that in that chapter we labeled the ordinate of the

distribution “density.” We also spoke in terms of intervals rather than in terms of specific

outcomes. Now we need to elaborate somewhat on those points.

Figure 5.2 shows the approximate distribution of the age at which children first learn to

walk (based on data from Hindley et al., 1966). The mean is approximately 14 months, the

standard deviation is approximately three months, and the distribution is positively skewed.

You will notice that in this figure the ordinate is labeled “density,” whereas in Figure 5.1 it

was labeled “relative frequency.” Densityis not synonymous with probability, and it is

probably best thought of as merely the height of the curve at different values of X. At the

same time, the fact that the curve is higher near 14 months than it is near 12 months tells us

that children are more likely to walk at around 14 months than at about one year. The rea-

son for changing the label on the ordinate is that we now are dealing with a continuous dis-

tribution rather than a discrete one. If you think about it for a moment, you will realize that

although the highest point of the curve is at 14 months, the probability that a child picked

at random will first walk at exactly14 months (i.e., 14.00000000 months) is infinitely

small—statisticians would argue that it is in fact 0. Similarly, the probability of first walking

at 14.00000001 months also is infinitely small. This suggests that it does not make any sense

to speak of the probability of any specificoutcome. On the other hand, we know that many

children start walking at approximately14 months, and it does make considerable sense

to speak of the probability of obtaining a score that falls within some specified interval.

Section 5.5 Probability Distributions for Continuous Variables 119

Density

Age (in months)

0 2468101214161820222426

Figure 5.2 Age at which a child first walks unaided

Density