0

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.

Rodolphe
  • 861
  • 5
  • 16
  • Are you really asking "what _is_ the variance of the sample mean, or of any other statistic, and where and how are they useful"? – Alecos Papadopoulos Dec 10 '14 at 13:38
  • Well... I think so, as the variance of any statistics is the squared statistics themselves. For example, the inverse of _sample_ variance is used in weighted least squares, but does it make sense to use the inverse of variance of the _sample mean_ instead ? – Rodolphe Dec 10 '14 at 13:46
  • Re "... cannot find any theory...": this is explained, in great detail, in several threads on this site. One that I could find is http://stats.stackexchange.com/questions/99284. Many more can be explored through the links at http://stats.stackexchange.com/search?q=weighted+least+squares+variance+minimize. – whuber Dec 10 '14 at 17:37

1 Answers1

1

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: $\quad SEM = \frac{SD}{\sqrt{n}} = \sqrt{\frac{\sum_i^n{(x_i - \bar{x})^2}}{n\, (n-1)}}$

Steven C. Howell
  • 391
  • 1
  • 16