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A statistic that serves as an estimator of the [[auto-correlation]] (auto-correlation function) of a [[Time series|time series]]. Let  $  x _ {1} \dots x _ {N} $
be a time series. The serial correlation coefficient of order  $  k $
is the statistic  $  r _ {k} $
defined by the formula

$$ \tag{* }
r _ {k}  = \ 

\frac{
\frac{1}{N - k }

\sum _ {i = 1 } ^ { N   - k } \{ \xi _ {i,k} \xi _ {i+k,k} \} }{\left [ 
\frac{1}{N - k }

\left \{ \sum _ {i = 1 } ^ { {N }  - k } \xi _ {i,k}  ^ {2} \right \}

\frac{1}{N - k }

\left \{ \sum _ {i = 1 } ^ { {N }  - k } \xi _ {i+k,k } ^ {2} \right \}
\right ]  ^ {1/2} }
 ,
$$

where

$$ 
\xi _ {i,k}  =  x _ {i} - {
\frac{1}{N - k }
 } \sum _ {i = 1 } ^ { {N }  - k }
x _ {i} .
$$

Statistics close to (*) but of a slightly simpler form are also used as serial correlation coefficients. The set of serial correlation coefficients is called a correlogram; this term is also used for the graph of  $  r _ {k} $
as a function of  $  k $.

Under various assumptions regarding the distribution of the  $  x _ {i} $, 
there are exact and approximate expressions for the distribution of the serial correlation coefficients, and of their moments. Serial correlation coefficients are used in statistical problems to discover the dependence of terms in a time series.

As well as the term  "serial correlation coefficient" , the term  "sampling auto-correlationsampling auto-correlation"  is used.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.M. Anderson,   "The statistical analysis of time series" , Wiley  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics" , '''3. Design and analysis, and time series''' , Griffin  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.J. Hannan,   "Time series analysis" , Methuen , London  (1960)</TD></TR></table>