Gini coefficient

“ Here, evidently, one of the conditions (166) – (169) must cease to hold true. Therefore, one of the differential coefficients formed by changing into in the first members of these conditions must have the value zero. (That it is the numerator and not the denominator in the differential coefficient which vanishes at the limit appears from the consideration that the denominator is in each case the differential of a quantity which is necessarily capable of progressive variation, so long at least as the phase is capable of variation at all under the conditions expressed by the subscript letters.) ”

“ The probability-coefficient for a generic phase will be times as great, viz., We shall say that such an ensemble as has been described is canonically distributed, and shall call the constant its modulus. It is evidently what we have called a grand ensemble. The petit ensembles of which it is composed are canonically distributed, according to the definitions of Chapter IV, since the expression
is constant for each petit ensemble. The grand ensemble, therefore, is in statistical equilibrium with respect to specific phases. ”

“ It is often convenient to represent in the form of a single differential coefficient, as
a block or matrix of ordinary differential coefficients. In this expression, may be a multiple quantity representing say independent variables, and another representing perhaps the same number of dependent variables. Then represents the differentials of the former, and the differentials of the latter. The whole expression represents an operator which turns into so that we may write identically Here we see a matrix of differential coefficients represented by a quotient. ”

“ The calculated value for the standard deviation of the frequency curve was while the observed was 1.039. The value of the standard deviation is really infinite, as the fourth moment coefficient is infinite, but as we have arbitrarily limited the infinite cases we may take as an approximation from which the value of the probable error given above is obtained. The fit of the curve is as follows:—
Comparison of Fit. ”

“ For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d 2u/dx2 is negative or positive. If d 2u/dx2 vanishes, then there is no maximum or minimum unless d 2u/dx2 vanishes, and there is a maximum or minimum according as d 4u/dx4 is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum. ”

“ The peculiar retrograding sequence of the numerical coefficients in Table IX, decreasing by 2 from katun to katun, as 2, 13, 11, 9, 7, 5, 3, 1, 12, etc., results directly from the number of days which the katun contains. Since the 13 possible numerical coefficients, 1 to 13, inclusive, succeed each other in endless repetition, 1 following immediately after 13, it is clear that in counting forward any given number from any given numerical coefficient, the resulting numerical coefficient will not be affected if we first deduct all the 13s possible from the number to be counted forward. ”

“ The correlation coefficient for Kemmerer's figures 1896-1908 is 83 per cent, which is far higher than that obtained by Persons for the period beginning in 1879.
25. For instance, no one would deny that the length and breadth of nuts are highly correlated. The coefficient of their correlation is 57 per cent. The height of a man and the breadth of his face are correlated to the extent of 35 per cent. ”

“ Bessel's Functions as Coefficients in an Expansion. — It is clear that eip°<«* = ei* or ew> si » * = g>» satisfy the differential equation (31) , hence if these exponentials be expanded in series of cosines and sines of multiples of 0, the coefficients must be Bessel's functions, which it is easy to see are of the first kind. To expand e'psin *_ put <â– <* = /, we have then to expand eM'-' -1 ) in powers of t. Multiplying together the two absolutely convergent series
we obtain for the coefficient of t m in the product P'" S . ”

“ Hence also, if the change in the external coördinates of an ensemble initially in statistical equilibrium consists in successive very small changes separated by very long intervals of time in which the disturbance of statistical equilibrium becomes sensibly effaced, the final diminution in the average index of probability will in general be negligible, although the total change in the external coördinates is large. ”

“ The curve of Frequency contains no easily perceived unit of variability like the Quartile of the Curve of Distribution. It is therefore not suited for and was not used as a first illustration, but the formula that expresses it is by far the more suitable of the two for calculation. Its unit of variability is what is called the ‘Standard Deviation,’ whose genesis will admit of illustration. How the calculations are made for finding its value is far beyond the reach of the present lessons. ”

“ We are speaking here, it must be understood, only of symmetrical curves: if there is asymmetry, i.e. if the Law of Error is different on different sides of the mean, a comparatively very small number of observations would suffice to detect the fact. But, granted symmetry and rapid decrease of frequency on each side of the mean, we could generally select some one species of the exponential curve which should pretty closely represent our statistics in the neighbourhood of the mean. That is, where the statistics are numerous we could secure agreement ”

“ Accordingly we seem to be justified in bringing to bear all 492 the same apparatus of calculation as in the former case. We take the initial average as the probable position of the true centre or ultimate average: we estimate the probability that we are within an assignable distance of the truth in so doing by calculating the ‘error of mean square’; and we appeal to this same element to determine the modulus, i.e. the amount of contraction or dispersion, of our derived curve of facility. ”

“ Again, if the value of a product of factors belonging to a system of units is multiplied by —1 when two factors change places, the same will be true of the product of any factors obtained by addition of the units. The kind of multiplication characterized by this relation and no other is called by Grassmann external or combinatorial. For our present purpose we may denote it by the sign It gives rise to multiple quantities of higher orders, units being required to express the products of two factors, units for products of three factors, etc. All products of more than factors are zero. ”

“ It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.
The impulse and real cause of inducement for the assumption of the group-transformation Gc is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation Gc. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group Ginfinity. ”