Page:Logic of Chance (1888).djvu/29

CHAPTER XIX.
THE THEORY OF THE AVERAGE AS A MEANS OF APPROXIMATION TO THE TRUTH.
§§ 1—4.	General indication of the problem: i.e. an inverse one requiring the previous consideration of a direct one.
	[I. The direct problem:—given the central value and law of dispersion of the single errors, to determine those of the averages. §§ 6—20.]
6.	(i) The law of dispersion may be determinate à priori,
7.	(ii) or experimentally, by statistics.
8, 9.	Thence to determine the modulus of the error curve.
10—14.	Numerical example to illustrate the nature and amount of the contraction of the modulus of the average-error curve.
15.	This curve is of the same general kind as that of the single errors;
16.	Equally symmetrical,
17, 18.	And more heaped up towards the centre.
19, 20.	Algebraic generalization of the foregoing results.
	[II. The inverse problem: given but a few of the errors to determine their centre and law, and thence to draw the above deductions. §§ 21—25.]
22, 23.	The actual calculations are the same as before,
24.	With the extra demand that we must determine how probable are the results.
25.	Summary.
	[III. Consideration of the same questions as applied to certain peculiar laws of error. §§ 26—37.]
26.	(i) All errors equally probable,
27. 28.	(ii) Certain peculiar laws of error.
29, 30.	Further analysis of the reasons for taking averages.
31—35.	Illustrative examples.
36, 37.	Curves with double centre and absence of symmetry.
38, 39.	Conclusion.