Statistical Methods for Psychology

To illustrate the binomial distribution we will take the classic example often referred to

as perception without awareness, or that loaded phrase “subliminal perception.”^3 A com-

mon example would be to flash either a letter or a number on a screen for a very short pe-

riod (e.g., 3 msecs) and ask the respondent to report which it was. If we flash the two

stimuli at equal rates, and if the respondent is purely guessing with a response bias, then

the probability of being correct on any one trial is .50.

Suppose that we present the stimulus 10 times, and suppose that our respondent was

correct 9 times and wrong 1 time. What is the probability of being correct 90% of the time

(out of 10 trials) if the respondent really cannot see the stimulus and is just guessing?

The probability of being correct on any one trial is denoted pand equals .50, whereas the

probability of being incorrect on any one trial is denoted qand also equals .50. Then we have

But so

Thus, the probability of making 9 correct choices out of 10 trials with p 5 .50 is remote,

occurring approximately 1 time out of every 100 replications of this experiment. This

would lead me to believe that even though the respondent does not perceive a particular

stimulus, he is sufficiently aware to guess correctly at better than chance levels.

As a second example, the probability of 6 correct choices out of 10 trials is the probabil-

ity of any one such outcome ( ) times the number of possible 6:4 outcomes ). Thus,

Here our respondent is not performing significantly better than chance.

Plotting Binomial Distributions

You will notice that the probability of six correct choices is greater than the probability of

nine of them. This is what we would expect, since we are assuming that our judge is oper-

ating at random and would be right about as often as he is wrong. If we were to calculate

=.2051

=

5040

24

(.00098)

=

10 # 9 # 8 # 7 # 6!

6! 4 # 3 # 2 # 1

(.5)^10

=

10!

6!4!

(.5)^6 (.5)^4

p(6)=

N!

X!(N 2 X)!

pXq(N^2 X)

p^6 q^4 C^106

=10(.001953)(.50)=.0098

p(9)=

10 # 9!

9 !1!

(.50^9 )(.50^1 )

10!= 10 # 9 # 8 #Á# 2 # 1 = 10 #9!

p(9)=

10!

9!1!

(.50^9 )(.50^1 )

p(X)=

N!

X!(N 2 X)!

pXq(N^2 X)

128 Chapter 5 Basic Concepts of Probability

(^3) Philip Merikle wrote an excellent entry in Kazdin’s Encyclopedia of Psychology (2000) covering subliminal per-
ception and debunking some of the extraordinary claims that are sometimes made about it. That chapter is avail-
able at http://watarts.uwaterloo.ca/~pmerikle/papers/SubliminalPerception.html.