Random Variable vs Statistic

A random variable turns the outcomes of a random process into numbers you can average and analyze. For example, if $X$ is the number of heads in two coin flips, then $X$ can equal 0, 1, or 2, each with its own probability. Random variables are discrete when their values are countable and continuous when they can take any value in an interval. Its long-run average is the expected value $E(X) = \sum x_i \, p_i$ (E of X, each value times its probability, summed).

A statistic varies from sample to sample because it depends on which individuals you happen to select. For example, if 52 out of 100 sampled voters favor a measure, the sample proportion is $\hat{p} = 0.52$ (p-hat, the estimate of the population proportion). Common statistics include the sample mean $\bar{x}$ (x-bar) and the sample standard deviation $s$. Its value changes with each new sample, which is what sampling distributions describe.