Educational Research - the Ethics and Aesthetics of Statistics

6 n = 1: The Science and Art of the Single Case in Educational Research 81
that describes the long-term stability of the mean of a random variable. Basically,
the law dictates that the more observations you make of a particular variable, the
more the sample mean will tend to approach and stay close to a particular value. If,
for example, we toss a (fair) coin just two or three times, then it is quite possible
that it will land as heads every time; if we continue for 10 or 20 tosses, then the proportion
of heads will start to approximate to 50%; and it will get closer and closer
to 50% the more times you toss the coin.
This law of large numbers tends to be extended (perhaps incorrectly, since it
is essentially a principle applicable to a random variable with a finite expected
value) as a general principle of research based on quantitative measure to indicate
that, other things being equal, the larger the sample, the closer to a true picture,
i.e. the more valid, will be the results. “The larger the sample size, the greater its
accuracy... The sampling error – the difference between the sample and the population
which are due for sampling – can be reduced by increasing sampling size”
[though] “after a certain level, increases in accuracy tend to trail off as sample size
increases” (Bryman & Cramer, 1990, p. 104). This principle applies in general to
both correlational research (e.g. To what extent is entry to higher education a function
of ethnicity, gender or social class?) and experimental research (How effective
are mentoring schemes aimed at under-represented groups in raising entry to higher
education in these groups?) In any research which is looking at the relationship
between different variables, for example, if there are very few observations, then
there are also few possible combinations of the values of the variables, and thus the
probability of obtaining by chance a combination of the values indicative of a strong
relation is relatively high. It is a feature of such research that when the enquiry has
to be based on a sample rather than the entire relevant population, we are nevertheless
interested in the confidence with which we can extrapolate from the sample to
that whole population – and, other things being equal, a larger sample reduces the
risk of error and or pure chance and increases the confidence with which one can
draw inferences from the sample to the whole population.
It is against these sorts of expectations from quantitative research traditions that
a sample size of a single case (n = 1), or even three or four cases can look faintly
ridiculous. But is it?
6.2 The Function of the Single Case Within a Quantitative
Research Tradition
It is worth noting, first of all, that the single case (and I shall use this also to include
a small number of cases such that would not normally be thought to be statistically
significant) can play an important and indeed devastating role even within the
broadly quantitative research tradition. I have thus far accepted without comment
the supposition which underpins the logic of a large part of empirical research in
education as elsewhere, that one can draw inferences from the an appropriately
constructed sample which has been the object of study to the wider population
from which it is selected: in other words, one can generalise from the particular