Statistical Analysis for Education and Psychology Researchers

estimation and inference. Typically the kinds of questions a researcher might want to ask
include:

‘How much confidence can we have that a sample frequency or

proportion represents the actual proportion in the population of interest?’

‘Does the outcome of interest, such as number of successes, number

surviving, pass rate or number of correct choices differ from what we

would expect by chance?’

‘Are the proportions of male and female truants in a school the same?’

Mean and Standard Deviation of a Binomial Variable

When a discrete random variable is a count of the successes in n independent trials which
each have the same probability of success, we use the term binomial variable. We
should realize that a binomial variable is a random variable or a statistic that is a count
and therefore has a sampling distribution or probability distribution. For every count
there is an associated probability. Just like distributions of variables described in Chapter
3 a binomial variable has a mean and a standard deviation. If we designate the count, X,
of a binomial variable from a binomial population, in notational form B(n, π), then the
mean of the binomial variable is given by,
μx=nπ
Mean
of a
Binomia
l
Variabl
e—4.2

and the standard deviation is,

(^) Standard
Deviation of
a Binomial
Variable—
4.3
When π is not known it is estimated from the sample data as the number of successes
divided by the number of trials. It is simply a proportion, P=X/n, where X is the count for
the binomial variable and n is the number of trials or observations.
Example 4.3: Mean and Standard Deviation of a Binomial
Variable
The probability of a secondary school having an unauthorized absence rate>1 % is about
π=0 5 This is knownaprioribased on Department for Education figures for the
Statistical analysis for education and psychology researchers 98