Clinical genetics / edited by Arnold Sorsby.

Date:
1953
Catalogue details

Licence: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)

Credit: Clinical genetics / edited by Arnold Sorsby. Source: Wellcome Collection.

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    SELECTED PEDIGREES It follows that sterilization of recessive homozygotes would be practically ineffective if the character is rare. More significant from the point of view of this chapter is that under this condition most of the recessive homozygotes would be born in marriages between heterozygotes. Calculation of the standard error In most cases we know Httle or nothing concerning the frequency of genetic characters of a population. If the data refer to a rather hmited population, the figure will, of course, be unrehable. It may be a little too high or too low at random. In order to evaluate the order of such inaccuracies it is usual to calculate the standard error. If the frequency is p per cent, the standard error e(p) will be equal to ^^1122 P)» where n is the number of individuals in the observed population. The formula shows that the size of the standard error is proportional to the square root of the number of individuals. Hence four times as many observations will be required to reduce the standard error by half. The reader is referred to ordinary text-books on statistics for the deduction of the formula (Dahlberg, 1940). It should be mentioned here only that the deduction is made with the aid of the theory of probability. Nevertheless, in order to illustrate the reasoning behind the formula, we shall illustrate the idea from the distribution obtained when a red or a black card is drawn from a pack of ordinary playing cards. If only one card is drawn, the probabihty of obtaining a red card is the same as the probability for a black card. If two cards are drawn (a card must be replaced in the pack before the next card is drawn), we can expect two red cards in one case, a black and a red card in two cases, and two black cards in one case. Continuing thus, we will obtain the following series of numbers: Combination 1 draw 1 + 1 total 2 draws ,, 2 draws 1+2 + 1 ,,2-4 ,, ,, 3 ,, 1+3 + 3 + 1 ,,3-8 ,, ,, 4,, 1+4 + 6 + 4 + 1 ,, 4-16 ,, 5 „ 1+5 + 10 + 10 + 5 + 1 „ 5-32 „ This pyramid of figures, which is called Pascal's Pyramid, is obtained with the aid of the binomial theorem. Each horizontal row consists of the coefficients in the expanded expression (a + b)^ for consecutive values of n. Thus: (a + ЪУ = la + lb (a + b)2 = la^ + 2ab + Ib^ (a + b)ä = la» + За^Ь + ЗаЬ^ + Ib^ (а + b)^ = la^ + 4a®b + 6a^b^ + 4ab® + Ib* (a + b)® = la® + 5a^b + lOa'^b^ + lOa^b» + 5ab^ + Ib® Such a distribution is called a binomial distribution. The same distribution can be obtained also with an apparatus which was designed by Francis Galton (Fig. 16). Balls are put in at the opening in the top and, striking the pegs standing out from the wall, fall down and collect in slots at the bottom (a form of the pinball gambHng machines). The distribution is seen in Fig. 17 which shows the paths the faUing balls may take. Let us suppose now that the apparatus is infinitely large and the balls infinite in number. Then the distribution would seem continuous, just as a polygon with an infinity of sides that is inscribed in a circle may be said to coincide with the circle. It is possible thus to calculate the curve 87