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The application of mathematical methods to results of observations, in order to form conclusions about the true values of unknown quantities. Any observational result arising in some way from measurements involves errors of various origins. Errors are divided into three groups: gross errors, systematic errors and random errors (concerning gross errors see [[Errors, theory of|Errors, theory of]]; in the rest of this article it will be assumed that the observations involve no gross errors). The result  $  Y $
of a measurement of some quantity  $  \mu $
is usually assumed to be a random variable; the measurement error  $  \delta = Y - \mu $
is then also a random variable. Let  $  b = {\mathsf E} \delta $
be its [[Mathematical expectation|mathematical expectation]]. Then

$$ 
Y  =  \mu + b + ( \delta - b).
$$

The quantity  $  b $
is called the systematic error, and  $  \delta - b $
the random error; the expectation of  $  \delta - b $
equals zero. The systematic error  $  b $
is frequently known in advance and is then easily eliminated. For example, in astronomy, when the angle between the direction of a star and the plane of the horizon is being measured, the systematic error is the sum of two errors: a systematic error originating in the instrument at the reading of the specific angle (the instrumental error) and a systematic error due to the refraction of light rays in the atmosphere. The instrumental error may be determined by consulting a correction table or graph for the instrument; the error due to refraction (for zenith distances less than  $  80 \circ $) 
may be calculated theoretically with adequate accuracy.

The effect of random errors is estimated by methods of the theory of errors. If  $  Y _ {1} \dots Y _ {n} $
are the results of  $  n $
independent measurements of  $  \mu $, 
carried out under identical conditions and by identical means, one usually puts

$$ \tag{1 }
\mu  \approx  \overline{Y}\; - b  = \ 

\frac{Y _ {1} + \dots + Y _ {n} }{n}
 - b,
$$

where  $  b $
is the systematic error.

If it is required to compute the value of a function  $  f ( y) $
at a point  $  y = \mu $, 
where  $  \mu $
is estimated on the basis of  $  n $
independent observations  $  Y _ {1} \dots Y _ {n} $, 
one approximates the required value by

$$ \tag{2 }
f ( \mu )  \approx  f ( \overline{Y}\; - b).
$$

Let  $  B $
be the expectation of

$$ 
\Delta  =  f ( \overline{Y}\; - b) - f ( \mu );
$$

then

$$ 
f ( \overline{Y}\; - b)  =  f ( \mu ) + B + ( \Delta - B).
$$

Therefore  $  B $
is the systematic error and  $  \Delta - B $
the random error of the approximation (2). If the random errors in the independent observations  $  Y _ {1} \dots Y _ {n} $
obey the same distribution and if the function  $  f ( y) $
is  "nearly"  linear in a neighbourhood of  $  y = \mu $, 
then  $  B \approx 0 $
and

$$ 
\Delta  \approx  f ^ { \prime } ( \mu ) \overline{ {( \delta - b) }}\; ,
$$

where  $  \overline{ {( \delta - b) }}\; $
is the [[Arithmetic mean|arithmetic mean]] of the random errors of the initial observations. This means that if

$$ 
{\mathsf E} ( \delta _ {i} - b)  ^ {2}  =  \sigma  ^ {2} ,\ \ 
i = 1 \dots n,
$$

then

$$ 
{\mathsf E} ( \Delta - B)  ^ {2}  \approx \ 
{\mathsf E} \Delta  ^ {2}  \approx \ 

\frac{[ f ^ { \prime } ( \mu )]  ^ {2} \sigma  ^ {2} }{n}
  \rightarrow  0
$$

as  $  n \rightarrow \infty $.

In the case of several unknown parameters, the observations are often processed using the method of least squares (cf. [[Least squares, method of|Least squares, method of]]).

If one is studying the dependence between two random variables  $  X $
and  $  Y $
on the basis of a sequence of  $  n $
independent observations, each of which is a vector  $  ( X _ {i} , Y _ {i} ) $, 
$  i = 1 \dots n $, 
subject to the (unknown) joint distribution of  $  X $
and  $  Y $, 
one uses the theory of [[Correlation|correlation]] to process the observations.

Whenever processing observations, one must make certain assumptions about the nature of the functional dependence, the distribution of the random errors, etc. It is therefore necessary to check the agreement between such assumptions and the results of the observations (both those actually used and others). See [[Statistical hypotheses, verification of|Statistical hypotheses, verification of]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Whittaker,   G. Robinson,   "The calculus of observations" , Blackie  (1944)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. Linnik,   "Methode der kleinste Quadraten in moderner Darstellung" , Deutsch. Verlag Wissenschaft.  (1961)  (Translated from Russian)</TD></TR></table>