>>13062785One way you can get a bound on the probability is with Chebyshev's inequality: for a random variable with finite mean and variance , the probability that takes a value deviating from by more than is For a sample of data, we can consider i.i.d. random variables , where and for all . Define the sample mean . It is easy to show that obeys and . Chebyshev's bound for then reads This doesn't give a probability of having a "particular" sample mean value, partly because that's a meaningless statement for continuous distributions. Instead it's a statement about how concentrated the sample mean is likely to be around the true mean. Equivalently, we can say that the probability that the sample mean lies in the range is at least .
