The essentials of mental measurement / by William Brown.

  • Brown, William, 1811-
Date:
1911
Catalogue details

Licence: In copyright

Credit: The essentials of mental measurement / by William Brown. Source: Wellcome Collection.

Provider: This material has been provided by King’s College London. The original may be consulted at King’s College London.

    142/172 (page 130)
    Since these equations are linear, our chance of a complex of variables lying between and f 1 + , ^, and ^, + 8^,,...^^ and f „ + g^,, Vn+i and Vn+i + 8r)n+u ... r),n and Vvi + 8r],n will be of the form Const. X e-k.-ka-kax BxM. ... X S77„+, ... S7;^...(ii). In this expression is a function of the f's only, q.^ is a function of rjn to t;^ only, and q^ is a function of the products of ^'s into 7}'s. In order to find the chance that our complex of variables lies between ^j, fi+S^i up to ^n, ^n + 8^n, we must integrate expression (ii) for all values of 77^+1 up to ijm between the limits + 00 and — 00 . Integrating with regard to 77,1^.1, we find that the expression keeps its general form, only rjn+i has disappeared. If now the process be repeated with 77^+2, the integration will again leave the expression of the same form except that 77,1+2 will have disappeared. Continuing this for an indefinite number of 77 integrations, all the 77's will at length disappear, and we shall be left with a quadratic function of ^'s only. In other w^ords, the chance of a complex of variables lying between and ^i + S^i, ^2 and ^2 + ^?2, ••• and ^n + S^n = Const. X - +P«»f«'+2^.af.f2+2(7.af.f3+ ■■■)d^4^, ...d^n-- -(iii). This expression represents the general form of the frequency surface connecting the x's. It may be noted that the fi*equency will be constant when 9u^i'+922^2'+ ■ ■ ■ +^nn^n'+25'i2^i?2+2sri3?i^3 + • • • =const... .(iv). This represents an ellipsoid in n-dimensional space. Taking the simplest case of two variables, expression (iii) becomes Oe-i(fi'.f.+'?»f»'+2fd^,d^^ (v). Integrating (v) for all values of from - 00 to + 00, we get the Gaussian curve of variation, so that 1 _ ^ 9^l\