Astronomers cannot avoid statistics, and there are several reasons
for this unfortunate situation. The most obvious is that every
observational science is one of *probabilities* - none more so than
astronomy, in which optical observers count individual photons from
faint objects until they have collected ``enough'', while their radio
colleagues persist with receivers generating noise signals of
amplitudes hundreds of times larger than those expected from faint
sources. We have all been taught by our Masters that no quantity
determined observationally is of use unless it has the proper error
associated with it; this implies that we know and understand both our
gear *and* some basic statistics. It also implies that other astronomers
are going to quote results in statistical terms - e.g. standard
errors, confidence limits - so that in self-defence, the implications
of these statistics must be familiar to us. Following the problems of
*detection* come the problems of *samples* - we are frequently
faced with
making deductions about various constituents of the Universe on the
basis of samples which are invariably small, and which are not easily
augmented. How can we convince ourselves/colleagues that an effect in
our sample indicates a Universal Truth? How likely is it that the
effect is only due to chance, to good/bad luck, to the First Law of
Experimentation ^{(1)} ?
We are not always aware that an appropriate test
exists. It *is* possible, for example, to test whether the ``degree of
woofliness'' (arbitrary and non-numerical scale) of the structures in a
sample of five radio sources is correlated with, say, galactic
latitude.

Practical problems such as this (?) receive little treatment in standard treatises on statistics (e.g. Kendall & Stuart (1), the definitive work). But most things necessary have been done by Those Who Have Gone Before, and some of their results are collected here, in this short series of articles which grew from lecture notes prepared for the new research students at MRAO.

The articles represent a personal point of view, and I make no claim for
originality or completeness. Results are presented with examples, but
with little justification and no theory, an (over) abundance of which
may be found in the standard works. Tables relevant to the material are
included, and these have been recomputed to avoid type-setting and
copyright problems. Throughout I emphasize two things: common sense, and
the necessity to use *non-parametric* methods. For the former there is no
substitute. The latter is forced upon us; the usual parametric tests
assume a knowledge of the underlying probability distributions of the
variables which we are sampling. (Examples are the Student *t*-test, the
*F*-test, and the standard correlation coefficient *r*, the
use of each
assuming Normal distributions for the variables involved.) We rarely
have this luxury, and, in fact, the underlying probability distribution
is frequently the object of our research. Therefore we must use
*non-parametric* tests. The books do not usually make the distinction and,
if they do, the non-parametric treatment is likely to comprise < 1 per
cent of the volume. The monograph by Siegel
(2) is an admirable
exception, and extensive reference is made to it later.

In these articles I shall discuss four topics which probably represent the most common entanglements of astronomers with statistics:

- Detection of signal; how was it done? Can you believe it?
- Correlation; does it exist? At what level of significance?
- Parameter estimation for a model; what are appropriate errors on the
parameters? Was the model/hypothesis reasonable in the first place?
- Comparison of samples: (a) with the predictions of a model/hypothesis, and (b) with each other; are they from the same population? Do they differ significantly? In some predicted sense?

As an unavoidable preamble I have to present some definitions and distributions (Section 2) which will certainly be encountered in the literature, both astronomical and statistical, and to discuss the Normal distribution (Section 3). Topic (I) is considered in Section 4, and the remaining topics in the subsequent articles.