Prognostic value of university entrance examinations in Scotland.

Date:
1934
Catalogue details

Licence: In copyright

Credit: Prognostic value of university entrance examinations in Scotland. Source: Wellcome Collection.

    135/140 (page 125)
    error of a mean (o,,) a text-book of statistics should be consulted. Suppose we know the Leaving Certificate marks in Mathematics of all candidates, and know that their mean is M and their standard deviation o. If samples of (say) 100 be taken at random (each being replaced before the next is taken), then their Means, 7, M2, Mg, etc., will not all be exactly equal to M, the mean of the whole; but they will not differ very much from it. If the differences M-m,, M-m,, M-mz, etc., be plotted, they will in general give a curve with a standard deviation of Ae, where n a is the standard deviation of the whole. This value (which has been written o,, in our tables) is the standard error of the means. Since in practice we do not know the standard deviation of the whole, but only that of a sample, it is customary to take the standard deviation of the sample as the best estimate of that of the whole. The standard error of the mean enables us to estimate the significance of the difference between two means. If two samples be taken of marks gained in a subject by the same students, their means will generally be only slightly different; if the samples be the marks of different students, the difference between their means may be great or small. The question arises: How great must the difference be before we can say with fair certainty that it indicates a real difference and that it is probably not the sort of difference that arises in random sampling? For practical purposes it can be taken that a differ- ence is significant if it be twice its standard error, for the odds are 20 to I against getting it by mere chance in samples of a homo- geneous population; if it be two and a half times its standard error the odds are 100 to 1. Now the standard error of a differ- ence between two means is 2 2 V/ Om, + Om, where o,,, and o,, are the standard errors of the means in question. In Table XX XIX the difference between the means of the two groups of Honours students (64:18 and 72-16) is 7-98, and its standard error is V 1-012 + 3-822, which is 3-95. 7-98 is about twice 3-95, hence the difference between the means of the two Honours groups is probably not an accidental one—in fact the odds against its being a chance difference are 20 to I.