THE LAW OF ACCIDENTAL ERRORS
Investigations of observations of various types show that accidental errors follow a definite law, the law of probability. This law defines the occurrence of errors and can be expressed in the form of equation which is used to compute the probable value or the probable precision of a quantity. The most important features of accidental errors which usually occur are:
(i) Small errors tend to be more frequent than the large ones; that is they are the most probable.
(ii) Positive and negative errors of the same size happen with equal
frequency ; that is, they are equally probable.
(iii) Large errors occur infrequently and are impossible.
PRINCIPLES OF LEAST SQUARES
It is found from the probability equation that the most probable values of a series of errors arising from observations of equal weight are those for which the sum of the squares is a minimum. The fundamental law of least squares is derived from this. According to the principle of least squares, the most probable value of an observed quantity available from a given set of observations is the one for which the sum of the squares of the residual errors is a minimum. When a quantity is being deduced from a series of observations, the residual errors will be the difference between the adopted value and the several observed values,
Let V1, V2, V3 etc. be the observed values x = most probable value