Statistical Analysis for Education and Psychology Researchers

area of all the bars would sum to 1 which is the sum of all the probabilities. If we extend
Pascal’s triangle to a sample of ten schools, the probability distribution is shown in Table
4.2

Table 4.2: Probability distribution for a binomial sample of n=10 with

p=0.5

Distribution 10A 9A1B 8A2B 7A3B 6A4B 5A5B 4A6B 3A7B 2A8B 1A9B 10B

Frequency 1 10 45 120 210 252 210 120 45 10 1

n above median 10 9 8 7 6 5 4 3 2 1 0

Probability 0.001 0.010 0.044 0.117 0.205 0.246 0.205 0.117 0.044 0.010 0.001

[(n+1) outcomes=11, and a total frequency of 2^10 = 1024 ]

To answer the question at the beginning of this example, ‘Could it happen by chance
that we obtain 3 schools in our sample that have above the national median pass rate?’
We can see from Table 4.2 that the probability of obtaining 3 schools above the median is
0.117, or approximately 12 per cent. This suggests that if we have chosen a random
sample then we would expect by chance, 12 schools in every 100 to lie above the
population median. The question now arises, ‘What is the critical probability level below
which we cannot accept that the outcomes would be expected to arise by chance alone?’
This brings us to the problem of statistical significance and p values. We will deal
with this when discussing hypothesis testing but for now we will make do with the
generally accepted convention that if we obtain a probability of less than 0.05,

written as p<0.05 (< means less than), then the results are deemed to be statistically
significant and not to have arisen by chance.
Since the observed probability p=0.117>p=0.05 we conclude that it is reasonable that
these data represent a random sample of schools. If we had observed either 8 or more
schools above the median, or 8 or more schools below the median, we would conclude
that it would have been very unlikely to have obtained this kind of distribution by chance.
We could then say that the schools did not represent a random sample of schools from the
population.

The Binomial Model

A general statistical model that describes the probability of the number of successes r in a
sample of size n is given by,
p=nCr×pr×( 1 −p)n−r
Binomial
Probability
—4.1

where: nCr=the binomial coefficient and generally refers to r successes in n trials (events).
Assume that we had selected another sample of 10 schools and found 7 had a pass rate
below the national median. Using the general notation, r=7, n=10, in notational form
nC
r=

(^10) C
7. To keep the notation simple, we will call below the median a success. Success
Statistical analysis for education and psychology researchers 96