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Investigations of observations of various types show that accidental errors follow a definite law, the law of probability.

**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 *V*1, *V*2, *V*3 etc. be the
observed values x = most probable value

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