Given a random variable, which is a set {x₁,...,x_n} of random numbers, we may want to know the arithmetic mean

<x> = (x₁ + ··· + x_n)/n,

the variance

v(x) = [(x₁ - <x>)² + ··· +(x_n - <x>)²]/n,

and the standard deviation (or root-mean-square deviation from the arithmetic mean),

σ(x) = √v(x).

σ(x) is an important measure of statistical dispersion.

Again, given n possible measurement outcomes v₁, ··· v_n with probabilities p_k = p(v_k), we have a probability distribution {p₁, ... ,p_n}, and we may want to know the expected value defined by

<x> = (p₁ x₁ + ··· + p_n x_n),

the variance defined by

v(x) = (p₁ x₁ - <x>)² + ··· +(p_n x_n - <x>)²,

and the standard deviation σ(x) = √v(x), which is a handy measure of the fuzziness of the observable whose possible values are the numbers {x₁,...,x_n}.

We have defined probability as a numerical measure of likelihood. So what is likelihood? What is probability apart from being a numerical measure? The frequentist definition covers some cases, the epistemic definition covers others, but which definition would cover all cases? It seems that probability is one of those concepts that are intuitively meaningful to us, but — just like time or the experience of purple — cannot be explained in terms of other concepts.