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:
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.
1 If the salami can fall out of the sandwich, it will. Back.