# Error in standard deviation and variance from error in data

I have a set of datapoints $x_i$ which have known upper bounds for absolute errors $\delta x_i$. (To clarify, this means each $x_i$ is actually $x_{i_0} \pm \delta x_i$). For simplicity, assume that all the errors are equal, i.e., $\delta x_i = \delta x$ $\forall x_i$.

The statistics of the datapoints is expected to fit a Gaussian. In other words, if one plots a histogram of the datapoints $x_i$, the histogram would fit a Gaussian reasonably well. Assume the mean of this Gaussian is zero.

How does one quantify the error in the standard deviation and variance of this Gaussian given the individual errors of the datapoints $x_i$ as described above?

• Can you clarify - you want to find the uncertainty in the s.d. and variance. (I.e. You aleady know how to estimate the s.d. and variance themselves)? – Rob Jeffries Nov 1 '14 at 21:33
• Would Cross Validated be a better home for this question? – Qmechanic Nov 1 '14 at 21:52
• @rob Jeffries, yes. I would like to calculate the uncertainty in the SD and variance, and not the SD and variance themselves. – eqb Nov 1 '14 at 23:33
• @Qmechanic, thanks. The datapoints originate from a physics experiment, so I thought to ask here. I'll post in CV as well. – eqb Nov 1 '14 at 23:49

If the data are normally distributed, then the variance of the variance is given by $$Var(s^2) = \frac{2 \sigma^4}{n-1},$$ where $\sigma$ is the standard deviation.
$$\sigma^2 = \frac{1}{n} \Sigma_i (x_i - \bar{x})^2$$
And $s^2$ is the unbiased sample variance, calculated from the data, where $s^2 = n \sigma^2/(n-1)$.
Hence the uncertainty in $s^2$ is approximately $\delta(s^2)=\sqrt{(2/n-1)}\sigma^2$. The uncertainty in $s$ is $\delta s = 0.5 \delta (s^2)/s$, which I think is $(2n)^{-1/2} \sigma$. (The factor of $\sqrt{2}$ looks suspicious to me).
Note that the standard deviation you should quote from your experiment is $s$, not $\sigma$.