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.