Statistical Analysis for Education and Psychology Researchers

median AA, the first school sampled is above the median and the second school sampled
is below the median AB, the first school is below the median and the second school is
above BA, and both the first and second schools are below the median, BB. If three
schools were sampled, the possible outcomes are shown in Figure 4.2.

Possibility: 1

with 3

above

3

with 2

above

3

with 1

above

1

with 0

above

=8

AAA AAB

ABA

BAA

ABB

BAB

BBA

BBB

Frequency of possibility: 3 3 1 =8

Probability: 1/8 3/8 3/8 1/8 =8

to 3 decimal places 0.125 0.375 0.375 0.125 =1

Figure 4.2: Pascal’s triangle of expected outcomes

If you look closely at Figure 4.2 you can see a pattern beginning to emerge. The total

number of outcomes for each school selected (event) is raised by a power each time you

move down a row. In the row for selection n=2, the middle number in the row 1 2 1 is 2.

This is obtained by summing (1+1) from the row immediately above, row n=1. This

pattern can be extended indefinitely.

You should note, for a sample of n, there are n+1 possible outcomes, i.e., if n =3 there

are 4 outcomes, 3 schools above the median, 2 schools above the median, 1 school above

the median and 0 schools above the median. Also, the probability of each possible

outcome is calculated by dividing the frequency of each outcome by the total number of

possible outcomes. The probability of all outcomes sums to 1.

The last line of Figure 4.2 are the values of the PROBABILITY DISTRIBUTION for

the binomial variable with n=3 and p=0.5. It is analogous to the relative frequency

distribution, if we replace relative frequencies with probabilities. This distribution could

be plotted as a bar chart with‘bars’ proportional in area to the probabilities The total

Probability and inference 95