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A statistic taking constant values on orbits generated by a group of one-to-one measurable transformations of the sample space. Thus, if  $  ( \mathfrak X , \mathfrak B ) $
is the sample space,  $  G = \{ g \} $
is a group of one-to-one  $  \mathfrak B $-
measurable transformations of  $  \mathfrak X $
onto itself and  $  t ( x) $
is an invariant statistic, then  $  t ( gx ) = t ( x) $
for all  $  x \in \mathfrak X $
and  $  g \in G $. 
Invariant statistics play an important role in the construction of invariant tests (cf. [[Invariant test|Invariant test]]; [[Invariance of a statistical procedure|Invariance of a statistical procedure]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Zacks,   "The theory of statistical inference" , Wiley  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.P. Klimov,   "Invariant inferences in statistics" , Moscow  (1973)  (In Russian)</TD></TR></table>