270 Unit 7 Critical reasoning: Advanced Level
simply because they are not proportions: they
are bald totals. If I supported my claim by
simply observing that there were 34 times as
many prison inmates in the USA as in
Germany, that would not be a false statement,
but it would be a misleading one in the
context of my argument. To compare the two
facts in any fair and meaningful way we need
the populations of the two countries as well as
the number of prisoners. The population of
the USA, as of 2011, was 312 million (in round
figures); that of Germany 82 million. We can
enter these numbers into a table, and calculate
the rates of imprisonment as follows:TABLE 1
Total
population
(millions)Prisoners Prisoners
per
100,000
population
Germany 82 67,000 82
USA 312 2,300,000 737The first two columns of the table contain the
(more or less) raw data; the third the processed
data. The processed data permits us to
compare like with like. We can now argue
legitimately that the proportion of the US
population that is in prison is around nine
times that of Germany: still a significant and
striking difference, but a long way short of 34
times! The difference may still fail to establish
that the number and length of prison
sentences are excessive. That remains a value
judgement, depending on what one means by
‘excessive’, and requiring rather more
information than we have in the table. But at
least the intermediate conclusion – the
contrast between the Germany and USA
rates – now has a firm evidential base.Selectivity
A second way in which data may mislead is
due to selectivity: choosing facts which suit a
theory or hypothesis and/or omitting those
which do not. One of the obvious weaknessesmultiplied, divided, rounded, converted into
percentages, plotted on graphs and so on.
Raw data is not necessarily altered by
processing – unless, of course, it is deliberately
falsified. Even so, the same data can be presented
in ways that support different inferences, some
perhaps more justified than others. It is how
statistics are used and presented therefore that
requires critical attention. As far as the raw
material is concerned we either believe it or we
don’t. (Grounds for believing or disbelieving a
claim were discussed in the chapters on
credibility in Unit 4.) But even if we believe the
data, and are satisfied with its accuracy, we may
still question the way it has been interpreted.
Like any argument, the premises can be true but
the reasoning still flawed. Statistical reasoning is
no different in this respect.
Here is a simple illustrative example. Take the
raw statistic that in 2010 there were 2.3 million
people in prison in the USA. (To be precise this
has already undergone some processing
because it has been rounded to the nearest
100,000, and presumably averaged over the
year. But within these bounds, it is either true or
false; we’ll assume it is true.) It is another fact
that in Germany the corresponding number
was a little over 67,000. These facts may come as
a surprise. They may prompt someone to argue
that the number of prisoners in the USA is
excessive or unnecessary, or inhumane, given
that the contrast is so striking between two
developed, and in many ways similar,
countries. But the numbers themselves do not
carry those implications. What is more, they
cannot be used in their raw form either to
strengthen or to weaken any such conclusion.‘Like with like’
One way in which statistics may mislead is by
comparing total numbers with proportions.
Suppose I did want to argue that the rate of
imprisonment in the USA was excessive, by
comparing it with that of another developed,
prosperous, democratic country. The two
figures above would be quite inadequate,