Chebyshev's Inequality and the Weak Law of Large Numbers

Chebyshev's inequality states that if are independent, identically distributed random variables (an iid sample) with common mean and common standard deviation and is the average of these random variables, then An immediate consequence of this is the weak law of large numbers, which states that as . 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.