For the past few years, stimulating research has been carried out on ordered statistics. Linear combinations of the sample ordered values are used to provide estimates of parameters of the population from which the sample is drawn. Such statistics are termed systematic by Mosteller. They are in common use, because they provide simple solutions of parametric problems of statistical estimation. Often they are inefficient, but this drawback is sometimes more than offset by the ease with which the calculations are performed.

Mosteller drew attention to the possible use of the statistics Wr , which he called quasi-range of rth order and defined as

where x 1, x 2,…, x n are ordered observations in a sample of n from a population f(x) (x 1 < x 2 < … < xn ). The distributions of quasiranges in samples drawn from a normal population have been studied by Godwin and Cadwell.

Current interest in world population and resources has resulted in the publication of several books in which the increase in world population over the last two thousand years has been traced or at least referred to. It may therefore be of interest to summarize some of the population statistics available for Greek and Roman times and the methods which have been used to obtain estimates of totals from them.

It is known that in many of the provinces of the Roman Empire a census was taken every 14 years which compares favourably in accuracy and completeness with those of modern times. Individual census returns have survived which show that in Egypt in the first two centuries A.D. the whole population was enumerated by households and by age, sex and occupation, but the only record of totals which have been preserved is a statement in Josephus's Jewish Wars to the effect that the population of Egypt excluding Alexandria was, according to the census returns, 7,500,000.

At the meeting of the Institute of Actuaries on 26 March 1956, it was brought out very clearly that the population problem for the world in general is very different from anything within the present experience of this country. In many areas the population is already outstripping the capacity for food production, and the rate of population increase, consequent upon improving mortality, gives cause for fears about the future.

One of the areas where this problem is becoming acute is the British Caribbean, where the island territories are already thickly populated, and there is no prospect that the rapidly increasing population can be adequately provided for. Lord Simon of Wythenshawe illustrated this very effectively in his 1954 report on the Population and Resources of Barbados. The report was printed privately, but the paper presented to the Institute described the position.

A table of for m = 1 (1) 20 and t = 1 (1) m was supplied by the writer in J.S.S. x (facing p. 64). The present table gives values of for m = 21 (1) 52 and t = 1 (1) n, where n = ½m if m is even, or ½(m — 1) if m is odd. By virtue of the identity I , it is unnecessary to tabulate the argument for higher values of t.

The table was constructed by means of the relation

,

and each column was checked by casting and using the relation

.