Statistical Methods for Psychology

For example, we might be interested in the probability that an infant will start walking at

14 months plus or minus one-half month. Such an interval is shown in Figure 5.3. If we ar-

bitrarily define the total area under the curve to be 1.00, then the shaded area in Figure 5.3

between points aand bwill be equal to the probability that an infant chosen at random will

begin walking at this time. Those of you who have had calculus will probably recognize

that if we knew the form of the equation that describes this distribution (i.e., if we knew

the equation for the curve), we would simply need to integrate the function over the inter-

val from ato b. For those of you who have not had calculus, it is sufficient to know that the

distributions with which we will work are adequately approximated by other distributions

that have already been tabled. In this book we will never integrate functions, but we will

often refer to tables of distributions. You have already had experience with this procedure

with regard to the normal distribution in Chapter 3.

We have just considered the area of Figure 5.3 between aand b, which is centered on

the mean. However, the same things could be said for any interval. In Figure 5.3 you can

also see the area that corresponds to the period that is one-half month on either side of

18 months (denoted as the shaded area between cand d). Although there is not enough

information in this example for us to calculate actual probabilities, it should be clear by in-

spection of Figure 5.3 that the one-month interval around 14 months has a higher probabil-

ity (greater shaded area) than the one-month interval around 18 months.

A good way to get a feel for areas under a curve is to take a piece of transparent graph

paper and lay it on top of the figure (or use a regular sheet of graph paper and hold the two up

to a light). If you count the number of squares that fall within a specified interval and divide

by the total number of squares under the whole curve, you will approximate the probability

that a randomly drawn score will fall within that interval. It should be obvious that the smaller

the size of the individual squares on the graph paper, the more accurate the approximation.

5.6 Permutations and Combinations

We will set continuous distributions aside until they are needed again in Chapter 7 and

beyond. For now, we will concentrate on two discrete distributions (the binomial and the

multinomial) that can be used to develop the chi-square test in Chapter 6. First we must

consider the concepts of permutations and combinations, which are required for a discus-

sion of those distributions.

120 Chapter 5 Basic Concepts of Probability

Density

Age (in months)

0

abcd

2468101214161820222426

Figure 5.3 Probability of first walking during four-week intervals centered on

14 and 18 months