Today I learnt that the standard deviation is calculated as square root of the mean of the squares of the deviations from the arithmetic mean of the distribution. The mean of the squares of the deviations from the arithmetic mean of the distribution is called the variance. The population standard deviation is noted as $\sigma$. However, no symbol was chosen for the variance. The population variance is simply noted as $\sigma^2$. The same thing applies for the sample standard deviation $s$ and the sample variance $s^2$. What is the reasoning for this? Are there any other examples like this?
(Compare this with the Hilbert space. Look at the section which says "An element A of B(H) is called 'self-adjoint' or 'Hermitian' if A* = A. If A is Hermitian and ⟨$Ax, x⟩ \ge 0$ for every x, then A is called 'nonnegative', written $A \ge 0$; if equality holds only when x = $\theta$, then A is called 'positive'. The set of self adjoint operators admits a partial order, in which $A \ge B$ if $A − B \ge 0$. If A has the form B * B for some B, then A is nonnegative; if B is invertible, then A is positive. A converse is also true in the sense that, for a non-negative operator A, there exists a unique non-negative square root B such that $A = B^2 = B * B$." Here, the non-negative operator is noted as A, and its non-negative square root is noted as B.)