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Normally a variance is calculated towards the mean ($\bar X_n$), so it looks like this:

$$ o^2_n = \frac{\sum_{i = 1} ^ n (X_i - \bar X_n)^2}{n} $$

What if we replace that mean by zero (even though the mean is not zero). What do you call this?

$$ o^2_n = \frac{\sum_{i = 1} ^ n (X_i)^2}{n} $$

Context: This formula is very useful for load balancing tasks as fairly as possible across workers/machines. The resulting number is higher if it is less fair. Unlike normal variance (or std dev), fairness focuses on the heaviest loaded worker/machine first.

And what do you call the square root of that formula?

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    I think you are after the [moment about $0$](https://en.wikipedia.org/wiki/Moment_(mathematics)). (Not aware of a standard name for the square root of a moment). – 2017-02-01
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    For a sample I don't know whether there's a standard name (by the way, for a sample you'd usually divide by $n-1$, not $n$). For a distribution the expected value of $X^2$ is also called the second moment of $X$. – 2017-02-01
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    Look up the distinction between 'central' and 'noncentral' moments. This distinction applies to both sample and population moments. – 2017-02-01

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I've seen it called the mean square. I don't know another name for it. Its square root is called the root mean square, or root-mean-square.

PS: Following "lulu" 's comment, it is of course also called the second moment. I think I missed that because I think of "second moment" as meaning $\operatorname{E}(X^2),$ and "second sample moment" as meaning $\sum_{i=1}^n X_i^2/n,$ where $X_1,\ldots,X_n$ is a sample rather than the whole population.