could be defined as any outcome of interest, such as count of deaths in dead/alive, count
of passes in pass/fail or count of yes in yes/no.
pr=probability of success, here 0.5, raised to the power of the number of
successes over all trials. In this example, pr=0.5^7
(1−p)n−r=1−probability of success in any trial, raised to the power of
number of trials less number of successes. In this example,
(1−p)n−r=0.5^10 −^3To answer the question ‘Could it happen by chance that we obtain 7 schools in our
sample that have below the national median pass rate?’ we need to recognize that the
underlying statistical model is a binomial model and evaluate formula 4.1:
(^10) C
7 ×0.5
(^7) ×0.5 10 − (^3)
The first term is a combinatorial, nCr, that is the number of combinations of n things
taken r at a time. This term is also known as the binomial coefficient. It is defined by,
where n! is n factorial which is the product of all integers from 1 to n (note 0!=1 as does
1!). We can now evaluate the binomial probability using formulae 4.1:
We can say that the probability of choosing, at random, 7 schools out of 10 below the
national median is 0.117. We can check this against the value derived from Pascal’s
triangle shown in Table 4.1.
Use of the Binomial Distribution in Research
Any data which is discrete and can be coded 0 or 1 such as success or failure, follow a
binomial distribution provided the underlying probability of a success, π, does not change
over the number of trials, n. Whereas the probability distribution is symmetric for a fixed
sample size when π=0.5, if π changes, for example, it reduces, the distribution also
changes shape (it would become positively skewed). Knowing the underlying distribution
of a discrete variable means we can estimate values and test hypotheses. Data in the form
of frequencies, proportions or percentages is very common in education and
psychological research and the variability of these sample statistics is very important in
Probability and inference 97