Not to be confused with standard deviation (SD).
Details #
Really understanding the difference between SD and SE took me quite a while early in teaching myself enough statistics to feel that I could understand most quantitative papers on our field. (And to get to the point where I could see when some of them misrepresent their findings as much more than they actually are!)
The difference is actually very simple but crucial to understanding statistical thinking. The SD (q.v.) is a description of the scatter of values of a variable in some dataset, the SE is not about the values themselves, it’s about some statistic summarising something about those values. In our field pretty much any SE you will see is going to be the SE of the mean: the SEM.
So if I have some scores on some measure, say 15 values:
1.3, 1.5, 2.5, 2.7, 2.3, 2.8, 1, 3.1, 2.3, 0.1, 2.1, 1, 3.7, 3.8 and 2.9
Their mean is 2.21 (to two decimal places) and the SD is 1.05. Those are summary descriptors of our little dataset, they are statistics as opposed to population parameters but, if the sampling that created that little dataset was random then we can see that sample mean as an estimate of the mean of some population from which the sample was taken. This is where the SE comes in as it is a indicator of how undercertain that mean is as a best guess of the population mean. For our little dataset the SE(mean) is 0.27. (For the mean, with random sampling, independent observations and assuming an infinitely large population then the formula for the SE is SD/sqrt(n) where n is the number of observations … and if the population distribution is Gaussian then it’s a good indicator of the uncertainty in estimating the population mean.
What does this mean? Well, perhaps rather confusingly, the SE is an estimate of the standard deviation of the means you would get if you kept generating samples. This next table shows the findings when I generated twenty samples, each of size 15. The columns show the means, SDs and SEs for each sample.
The SD of those twenty means is .279 which is lower than the mean of those SEs. That is because I modelled the scores as a flat distribution between 0 and 4. When I did a similar simulation but sampling from a Gaussian distribution the SD of the means was .238 and mean of the SEs was .255, a much closer fit.
I guess that just goes to show that things working properly does depend on the distributions fitting the assumed ones, so often the assumption that the population distribution is Gaussian. (That’s the case when using the sample SEM as an estimate of the SD of the means across repeated samples.)
As the SEM is the SD divided by the square root of the sample size (n) it will get smaller, for similar SDs, the bigger your n. That’s what we’d expect as larger samples, assuming that they are fairly randomly created, are going to be more precise estimates of the population parameter.
But still, so what?! The crucial thing is that the SE for many sample statistics, including the sample mean allows us to compute confidence intervals that tell us how precisely we might have estimated a population parameter (i.e. the unknowable population mean). But confidence intervals are for another glossary entry.
Try also #
- (Arithmetic) mean
- Confidence intervals (CIs)
- Estimate, estimation
- Gaussian (“Normal”) distribution
- Independence of observations
- Sample
- Sampling and sample frames
- Square root
- Standard deviation (SD)
- Variance: introduction
- Variance: computation and bias
Chapters #
Not covered in the OMbook but confidence intervals and the principle that the larger the sample you have, other things being equal, the better you can estimate population values runs through the entire book as it’s a foundational building block of understanding statistics.
Online resources #
None yet.
Dates #
First created 22.iii.25.
