= Solution
In the <high-resolution quantization approximation>, a smooth input distribution varies little across one conversion interval, so <rounding> to the nearest level gives an approximately <uniform quantization error> $\epsilon\in[-\Delta V/2,\Delta V/2]$. Therefore
$$
\langle\epsilon\rangle=0,\qquad
\langle\epsilon^2\rangle=\frac1{\Delta V}\int_{-\Delta V/2}^{\Delta V/2}\epsilon^2\,d\epsilon
=\frac{\Delta V^2}{12},\qquad
\boxed{\epsilon_{\rm rms}=\frac{\Delta V}{\sqrt{12}}}.
$$
The input must stay within the <analogue-to-digital converter>'s range. A large <standard deviation> alone does not guarantee <uniform quantization error>: an input concentrated exactly on widely separated conversion levels has zero error. Truncation without a half-step correction also has a different mean and <root mean square>.
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