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A Statistician: is ``a man prepared to estimate the probability that the Sun will rise tomorrow in the light of its past performance'' (3). This definition, which I admit to taking out of context, is by an eminent statistician; it may confirm our worst suspicions about statisticians eminent or otherwise. At the very least it should emphasize care of definition - what is tomorrow when the Sun does not rise? Let us be more careful:

``Statistics'' is a term loosely used to describe both the science and the values. In fact, the science is

Statistical Inference: the determination of properties of the population from the sample,


Statistics are values (usually, but not necessarily, numerical) determined from some or all of the values of a sample.

Good statistics are those from which our conclusions concerning the population are stable from sample to sample, while good samples provide good statistics, and require appropriate design of experiment.

The ``goodness'' of the experiment, the sample, or the statistic is indicated by the

Level of Significance: suppose we perform an experiment to distinguish between two rival hypotheses, the null hypothesis (H0; ``failure'', no result, no detection, no correlation) and its alternative (H0; ``success'', etc.). Before the experiment we make ourselves very familiar with ``failure'' by determining, assuming H0 to be true, the set of all possible values of the statistic under test, the statistic we have chosen to use in deciding between H0 and H1. Furthermore, suppose that when we do the experiment we obtain a value of the statistic which is ``unusual'' in comparison with this set, so unusual that, say, only 1 per cent of all values computed under the H0 hypothesis are so extreme. We can then reject H0 in favour of H1 at the 1 per cent level of significance. The level of significance is thus the probability of rejecting H0 when it is, in fact, true.

Now consider N values of xi where i = 1, 2 . . . N and x may have a continuous or a discrete distribution. The following definitions are general:

1. Location measures

Equation 1

Median: arrange xi according to size; renumber. Then
xmed = xj where j = N / 2 + 0.5, N odd
= 1/2 (xj + xj+1 where j = N / 2, N even.

Mode: xmode is the value of xi occurring most frequently.

2. Dispersion measures

Equation 2

Equation 3

Equation 4

3. Moments

Equation 5

(Moments may be taken about any value of x; those about the arithmetic mean as above are termed central moments.) Note that µ2 = sigma2; this and the next few moments characterize a probability distribution (defined below). The first two moments are useless, since µ0 ident 1 and µ ident 0.

Skewness: beta1 = µ32 / µ23 indicates deviation from symmetry;
= 0 for symmetry about µ.

Kurtosis: beta2 = µ4 / µ22 indicates degree of peakiness;
= 3 for Normal distribution.

Finally, consider probability distributions: if x is a continuous random variable, then f (x) is its probability density function if it meets these conditions:

(1) Probability Equation 6

(2) Equation 7


(3) f (x) is a single-valued non-negative number for all real x. The corresponding distribution function is

Equation 8

Table 1. The common probability distributions.

Distribution Density function Mean Variance Raison d'Être

Uniform (a+b)/2 (b-a)/12

In the study of rounding errors, and as a tool in theoretical studies of other continuous distributions.

Binomial np npq

x is the number of ``successes'' in an experiment with two possible outcomes, one (``success'') of probability p, and the other (``failure'') of probability q = 1 - p. Becomes a Normal distribution as n -> infty.

Poisson µ µ

The limit for the Binomial distribution as p << 1, setting µ ident np. It is the ``count-rate'' distribution, e.g. take a star from which an average of µ photons are received per Deltat (out of a total of n emitted; p << 1); the probability of receiving x photons in a Deltat is f (x;µ). Tends to the Normal distribution as µ -> infty.

Normal (Gaussian) µ sigma2

The essential distribution; see text. Central Limit Theorem ensures that majority of ``scattered things'' are dispersed according to f (x;µ,sigma).

Chi-square nu 2nu

Vital in the comparison of samples, model testing; characterizes the dispersion of observed samples from the expected dispersion, because if xi is a sample of nu variables Normally and independently distributed with means µi and variances sigmai, then obeys f (chi2; nu) Invariably tabulated and used in integral form. Tends to Normal distribution as nu -> infty.

Student t 0 nu/(nu-2)
(for nu > 2)

For comparison of means, Normally-distributed populations; if n xis are taken from a Normal population (µ,sigma), and if xs and sigmas are found as in text, then t = is distributed as f (t,nu) where ``degrees of freedom'' nu = n - 1. Statistic t can also be formulated to compare means for samples from Normal populations with same sigma, different µ (4). Tends to Normal as nu -> infty.

(for nu2 > 2)

For comparison of two variances, or of more than two means; if two statistics (chi1 and chi2) each follow the Chi-square distribution, then is distributed as f(F;nu1,nu2). Care required in application; see (4), (9).

Probability densities and distribution functions may be similarly defined for sets of discrete values x = x1, x2. . . .xn, and for multivariate distributions. The better-known (continuous) functions appear in Table 1, together with location and dispersion measures - note that the previous definitions for these may be written in integral form for continuous distributions. The table includes some indication of how and/or where each distribution arises, and for most of them, I avoid further discussion. But there is one whose rôle is so fundamental that it cannot be treated in such a cavalier manner. This follows in the next section.

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