Why some calculations noted as "sym^2" and "sym", while others noted as "symA" and "symB", where "symB" is the square root of "symA"?

Question

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.)

Answer 1

The reason is purely historical. When Karl Pearson introduced the standard deviation he used the symbol $\sigma$ and then when Fisher wanted a symbol for its square he used $\sigma^2$ but introduced a new name: variance. This can be found in https://mathshistory.st-andrews.ac.uk/Miller/mathsym/stat/ several screens down the page. Note that there are other cases in statistics where we use a squared quantity because it seems a natural way to express things: $\chi^2$, and Hotelling's $T^2$ for example although there are other examples.

In fact I would question your assertion that there is no notation for the variance as many people use Var() or var() for it even though there is a possible confusion with variable.

Just to add further issues the sample standard deviation is denoted $s$ but the definition you give is for the population standard deviation which is indeed $\sigma$.

The reference given above was provided by user Gareth Rees in a comment on MSE.