Chebyshev's inequality states that if
are independent, identically distributed random variables (an iid sample) with common mean
and common standard deviation
is the average of these random variables, then
An immediate consequence of this is the weak law of large numbers, which states that
. The blue dots in the image are the means of 100 different iid samples. In this Demonstration, these samples are drawn from a normal distribution with mean and standard deviation controlled by the top two sliders, but both of these results hold for any underlying distribution (with finite mean). The two red lines mark the endpoints of the interval (
). The dashed line marks the location of the mean
. The fraction of blue dots outside these lines will usually be smaller than the theoretical upper bound given in Chebyshev's inequality—in many instances this bound is very crude.