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 determinable _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._

THE LOGIC OF CHANCE.