Page:EB1911 - Volume 22.djvu/415

table the columns for the normal distribution and for the discrepancy e should each be halved; and accordingly the column for e²/m should be halved. Thus e²/m being reduced to 22.9, P as found from Professor Pearson's table is between 995 and 629. That is, such a distribution might be expected to occur once on an average some once or twice in a hundred times. If actual duplication of this sort is not common in statistics,^[1] yet in all such applications of the Pearsonian criterion—and in other calculations involving the number of observations, in particular the determinations of probable error—a good margin is to be left for the possibility that the n observations are not perfectly independent: e.g. the accidents of wind or nerve which affected one shot may have affected other shots immediately before or after.

Belt.	Observed  Frequency.	Normal  Distribution.	e.	e2/m.
1	1	1	0	0
2	4	6	− 2	0.667
3	10	27	−17	10.704
4	89	67	+22	7.224
5	190	162	+28	4.839
6	212	242	−30	3.719
7	204	240	−36	5.400
8	193	157	+36	8.255
9	79	70	+ 9	1.157
10	16	26	−10	3.846
11	2	2	0	0
	1000	1000	—	45.811

158. (2) The Generalized Law of Error.—That the normal law of error should not be exactly fulfilled is not disconcerting to those who ground the law upon the plurality of independent causes. On that view the normal law would only be exact when the numbers of elements from which it is generated is very great. In general, when that number is large, but not indefinitely great,^[2] there is required a correction owing to one or other of the following imperfections: that the elements do not fluctuate according to the normal law of frequency; that their fluctuations are not independent of each other; that the function whereby they are aggregated is not linear. The correction is formed by a series of terms descending in the order of magnitude.

159. The first term of this series may be written

−2(k/c³)[x/c − 2x³/3c³];

where c²/2 is the mean square of deviation for the compound and also the sum of the mean squares of deviations for the component Second and Third Approximations. elements, k₁ is the mean cube of deviations for the compound and the sum of the mean cubes for the components, and the elements are supposed to be such and so numerous that k₁/c³ is of the order 1/√n. This second approximation, first given by Poisson, was rediscovered by De Forest.^[3] The present writer has obtained it^[4] by a variety of methods. By a further extension of these methods a third and further approximations may be found. The corrected normal law is then of the form^[5]

$z={\frac {1}{\sqrt {\pi {\text{c}}}}}\left(\exp {-{\frac {x^{2}}{{\text{c}}^{2}}}}\right)\left[1-2{\text{k}}\left({\frac {x}{\text{c}}}-{\frac {2x^{3}}{3c^{3}}}\right)+{\text{k}}^{2}\left(-{\frac {5}{3}}+10{\frac {x^{2}}{c^{2}}}-{\frac {20}{3}}{\frac {x^{4}}{{\text{c}}^{4}}}+{\frac {8}{9}}{\frac {x^{6}}{{\text{c}}^{6}}}\right)+{\text{k}}_{2}\left({\frac {1}{2}}-2{\frac {x^{2}}{{\text{c}}^{2}}}+{\frac {2}{3}}{\frac {x^{4}}{{\text{c}}^{4}}}\right)\right]$;

where k = k₁/c³, k₂ = k₂/c⁴, k₁ and c are defined as above, k₂ is the sum of the respective differences for each element between its mean fourth power of error and thrice its mean square of error,^[6] and also the corresponding difference for the compound. The formula may be verified by the case of the binomial, considered as a simple case of the law of great numbers. Here

c² = 2npq, k₁ = npq(q − p), k₂ = npq(1 − 6pq).^[7]

These values being substituted for the coefficients in the general formula, there results an expression which may be obtained directly by continuing^[8] to expand the expression for a term of the binomial.

In virtue of the second approximation a set of observations is not to be excluded from the affinity to the normal curve because, like the curve of barometric heights,^[9] it is slightly asymmetrical. In virtue of the third approximation it is not excluded because, like the group of shot-marks above examined, it is, though almost perfectly symmetrical, in other respects apparently somewhat abnormal.

160. If the third approximation is not satisfactory there is still available a fourth, or a still higher degree of approximation.^[10] Higher Approximations. The general expression for y which (multiplied by ∆x) represents the probability that an error will occur at a particular point (within a particular small interval) may be written

$e^{-k_{1}{\frac {1}{3!}}\left({\frac {\text{d}}{{\text{d}}x}}\right)^{3}+k_{2}{\frac {1}{4!}}\left({\frac {\text{d}}{{\text{d}}x}}\right)^{4}-\cdot \cdot \cdot +(-1)^{t}k_{t}{\frac {1}{(t+2)!}}\left({\frac {\text{d}}{{\text{d}}x}}\right)^{t+2}\cdot \cdot \cdot }y_{0}$,

where y₀ is (the normal error-function) 1/√(2kπ)e^−x2/2k, k is the mean square of deviation; k₁, k₂, . . ., &c., are coefficients formed from the mean powers of deviation according to the rule that k_t is the difference between the tth mean power as it actually is and what it would be if the (t−1)th approximation were perfectly correct. Thus k₁ is the difference between the actual mean third power and what the third power would be if the first approximation, the normal law, were perfectly correct, that is, the difference between the actual mean third power, often written μ₃, and zero, that is μ₃. Similarly k₂ is the difference between the actual mean fourth power of deviation, say μ₄, and what that mean power would be if the second approximation were perfectly correct, viz. 3k². Thus k₂ = μ₃ − 3k². The series k₁, k₃, k₅, &c., k, k₂, k₄, &c., form each a succession of terms descending in the order of magnitude, when each k, e.g. k_t, has been divided by the corresponding power, i.e. the power (t+2) of the parameter or modulus c = √(2k), which division is secured by the successive differentiations of y₀, with which each k is associated, e.g. k_t with $\left({\frac {d}{dx}}\right)^{t+2}$. Moreover, the first term of the odd series of k's when divided by the proper power of the parameter, viz. c³ is small in comparison with the first term of the even series, viz. k, properly referred—divided by c² ( = 2k).

161. Whatever the degree of approximation employed, it is to be remembered that the law in general is only applicable to a certain Character of the Approximation. range of the compound magnitude here represented by the abscissa x.^[11] The curve of error, even when generalized as here proposed, coincides only with the central portion—the body, as distinguished from the extremities—of the actual locus; a greater or less proportion.

162. The law thus generalized may be extended, with similar reservations, to two or more dimensions. For example, the second approximation in two dimensions may be written Extension to Two or More Dimensions.

$z_{0}-{\frac {1}{3!}}\left({_{3,0}}{\text{k}}{\frac {d^{3}{\text{z}}_{0}}{d{\text{x}}^{3}}}+3{_{2,1}}{\text{k}}{\frac {d^{3}{\text{z}}_{0}}{d{\text{x}}^{2}d{\text{y}}}}+3{_{1,2}}{\text{k}}{\frac {d^{3}{\text{z}}_{0}}{d{\text{x}}d{\text{y}}^{2}}}+{_{0,3}}{\text{k}}{\frac {d^{3}{\text{z}}_{0}}{d{\text{y}}_{0}}}\right)$;

where z₀ is (the normal error-function)

1/π(1 − r²)exp.−(x² - 2rxy + y²)/(1 − r²),

x and y are (as before) co-ordinates measured from the centre of gravity of the group as origin, each referred to (divided by) its proper modulus; r is the ordinary coefficient of regression; _3,0k is the mean value of the cubes x³, _2,1k is the mean value of the products x²y, and so on; all these k's being quantities of an order less than unity. This form lends itself readily to the determination of a second approximation to the regression-curve, which is the locus of that y, which is the most probable value of the ordinate corresponding to an assigned value of x. Form the logarithm of the above-written expression (for the frequency-surface); and differentiate that logarithm with respect to x. The required locus is given by equating this

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1. It is frequent in the statistics of wages.
2. See on this subject, in addition to the paper on the “Law of Error” already cited (Camb. Phil. Trans., 1905). another paper by the present writer, on “The Generalized Law of Error,” in the Journ. Stat. Soc. (September, 1906).
3. The Analyst (Iowa), vol. ix.
4. Phil. Mag. (Feb., 1896) and Camb. Phil. Trans. (1905).
5. The part of the third approximation affected with k² may be found by proceeding to another step in the method described (Phil. Mag., 1896, p. 96). The remaining part of the third approximation is found by the same method (or the variant on p. 97) from the new partial differential equation dy/di = 1/24d⁴y/dx⁴, where k₂, is the difference between the actual mean fourth power of deviation and what it would be if the normal law held good. Further approximations may be obtained on the same principle.
6. μ₄ − 3μ₂² in the notation which Professor Pearson has made familiar.
7. Cf. Pearson, Trans. Roy. Soc. (1895), A, clxxxvi. 347.
8. Above, § 103, referring to Todhunter, History, art. 993. The third (or second additional term of) approximation for the binomial, given explicitly by Professor Pearson, Trans. Roy. Soc. (1895), A, footnote of p. 347, will be found to agree with the general formula above given, when it is observed that the correction affecting the absolute term, his y₀, disappears in his formula by division.
9. Journ. Stat. Soc. (1899), p. 550, referring to Pearson, Trans. Roy. Soc. (1898), A.
10. Practically no doubt the law is not available beyond the third or fourth approximation, for a reason given by Pearson, with reference to his generalized probability-curve, that the probable error incident to the determination of the higher moments becomes very great.
11. This consideration does not present the determination of the true moments from the complete set of observations if homogeneous, according as the system of elements fulfils more or less perfectly certain conditions.