In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode.
Winkler in 1866 extended Gauss's inequality to rth moments [1] where r > 0 and the distribution is unimodal with a mode of zero.
[2][3] Gauss's bound has been subsequently sharpened and extended to apply to departures from the mean rather than the mode due to the Vysochanskiï–Petunin inequality.
The latter has been extended by Dharmadhikari and Joag-Dev[4] where s is a constant satisfying both s > r + 1 and s(s − r − 1) = rr and r > 0.
It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions.