Squared uncertainties: what may it be used for? If the squared standard deviation of a set of values is the variance of this sample, then, what is the squared standard error of the mean of this sample ? and what may it be used for ?
A quick search through the litterature showed that the inverse of squared standard error of the mean has been used sometimes as weights in weighted least square linear modeling, but I can't find theory backing it up.
This question also applies to x% confidence intervals, derived from the standard error of the mean and to uncertainties in general.
 A: This article, specifically in the context of biomedicine, provides a clear introduction and comparison of variance, standard deviation, and standard error of the mean.
In short, the squared standard error of the mean (SEM) is the variance divided by the sample size: $SEM^2 = \dfrac{\sigma}{n}$
Here is a summary motivating where this comes from.  When considering a complete population you have,


*

*Mean of a population: $\quad \mu = \frac{\sum_i^N x_i}{N}$

*Variance of a population: $\quad \sigma^2 = \frac{\sum_i^N{(x_i - \mu)^2}}{N}$

*Standard deviation of a population: $\quad \sigma = \sqrt{\frac{\sum_i^N{(x_i - \mu)^2}}{N}}$

*Number in Population: $\quad N$
When considering a sample from the population this becomes (note these are estimators of the true population statistic based on a random sample from the population),


*

*Estimate of the population mean from a sample: $\quad \bar{x} = \frac{\sum_i^n x_i}{n}$

*Estimate of the population standard deviation from a sample: $\quad SD = \sqrt{\frac{\sum_i^n{(x_i - \bar{x})^2}}{n-1}}$

*Number in the sample: $\quad n$

*Standard error of the mean: $\quad SEM = \frac{SD}{\sqrt{n}} = \sqrt{\frac{\sum_i^n{(x_i - \bar{x})^2}}{n\, (n-1)}}$
