Why is the RMS value taken to calculate uncertainty in random errors My Text Book mentions the use of RMS (Root Mean Square) to calculate the value of uncertainty.
"Random errors are handled using statistical analysis. Assume that a large number ($N$) of measurements are taken of a quantity $Q$ giving values $$Q_1, Q_2, Q_3,…Q_N.$$ Let $Q$ be the mean value of these measurements.
$$\langle Q\rangle = \sum_{i=1}^n \frac{Q_i}{N}  $$
and let $d$ be the deviation in the measurements,
$$d=\sqrt\frac{\sum_{i=1}^n(Q-\langle Q\rangle)^2}{N}$$
The result of the measurement is quoted (assuming systematic errors have been
eliminated) as,
$$Q = \langle Q\rangle +d$$
My question is, how did we arrive at the expression for $d$ in the first place?
 A: Perhaps two examples will clarify things

*

*Data 1,2,3

The mean is $$<Q> = \sum_{i=1}^n \frac{Q_i}{N} = \frac{6}{3} = 2$$
The standard deviation formula is a way to calculate a 'measure of spread' i.e. an indication of how spread out the data is.  This can also be an indication of the errors in the measurement.
Since some data is lower than the mean and some higher, the square is included before the square root to make all the differences from the mean positive, otherwise we might be using a formula that gives a spread of zero.
$$d=\sqrt\frac{\sum_{i=1}^n(Q-<Q>)^2}{N} =\sqrt\frac{(-1)^2+0^2+(1)^2}{3} = 0.8165$$


*Data 10,20,30

The mean is now $20$, and $d$ works out to be $8.165$
The second set of data is more spread out, or the error in the measurements is higher.
In each case the 'true' value of the thing being measured is likely to be in the range $Q = <Q> \pm d$
In the second example, if a theory predicted that $Q=40$ and measurements had given $20 \pm 8.165$ the probability that  the $40$ could be true, can then be found using a normal distribution and finding the probability that a measurement would be  $\frac{20}{8.165}$ i.e. $2.45$ standard deviations  or $2.45d$ from the mean.
If the true standard deviation isn't known, but is estimated from the data, there is a further adjustment, but that's a technicality...
A: When the errors that contribute to fluctuations in a measured value are normal in their distribution, an infinite number of measurements will yield a Gaussian distribution. The mean $\mu$ and standard deviation $\sigma$ (square root of the variance) characterize the distribution and its uncertainty.
For a finite number of measurements on values $V_j$ where contributions to error in $V_j$ remain normal in their distribution, we use the average $\langle V \rangle$ and standard uncertainty $\Delta V$ of the distribution to define the expected value and its overall uncertainty. This is the report that you present.
The average approaches the mean $\langle V \rangle \rightarrow \mu$ and the standard uncertainty approaches the standard deviation $\Delta V \rightarrow \sigma$ as the number of measurements we make approach infinity. We can characterize the approach using the standard uncertainty of the mean $S_m$. Further considerations lead to approaches such as the  t-test for the confidence of the measured average relative to the true mean.
When the errors that contribute to $V_j$ are not normal in their distribution, additional complexities arise. Also, when we have to combine various values using an equation, we have to consider how we must propagate the uncertainties in the measured values. Both topics are expansions of your starting question.
A thorough reference to appreciate uncertainties or errors in measurements can be found in the Guide to Uncertainties in Measurements (GUM).
