Standard error (SE) measures the variation in the means from multiple sets of measurements, e.g. mean of the means. It indicates how much the sample mean is likely to vary from the true population mean.
Note that, by concept, it measures stats from multiple measurement, but we can estimate the standard error even by a single set.
Differences between standard error and standard deviation
Standard deviation (SD) quantifies the variation within a single set of measurements, while Standard error quantifies the variation of the mean of the means.
Calculate standard error of the mean
Take multiple samples
Calcualte the mean for each sample
Calculate the standard deviation of the means = standard error (SE)
Due to Central limit theorem, the standard error is less than standard deviation and will be leaned towards center.
Calculate standard error of standard deviation
Similar to above, we can take the SD from each sample, and calculate the SD of SDs, which will be the standard error of standard deviations. This tells us how the standard deviations of multiple samples are dispersed.
The similar steps can be applied to other statistic like means, percentiles, etc. It’s called standard error of the X.
The easy way to estimate standard error
- Bootstrapping can be used in many cases, which uses computer to do the trick
- In rare cases, we can use a formula to estimate SE like SE of the mean