SD shows the variation eg caused by biological variation (ie natural) in the data... whereas the SEM gives an indication of the precision of the estimate of the mean (similar to a confidence interval). I always show SEM - since most journals use this etc, SD bars are always way bigger! the bigger your sample size, the smaller the SEM and so the more sure you can be of your estimate of the mean.
I thought I had a handle on this but the more I think about it the more confused I get! personally if reporting stats I'd put in the SD but if displaying on a graph I'd use SE for error bars. Why? I don't really Know that's just what I'd do!
SD gives an idea of the spread of the data. Whilst SE also does the SE also takes into account the number of observations you have, as the greater n you have the smaller the SE. This is not the case with SD.
In some cases it makes sense to use SD. For example if just talking about your sample, such as reporting the age of participants, wheras I think SE relates more to the population - There is a 95% chance the actual mean will fall 1.96*SE from your mean value. Thus it gives an idea of how representative your mean is likely to be of the general popluation.
As for why report SE with t-test, I don't really know. I guess as the t-test measures whether the means are different or whether the error bars overlap, by reporting SE and the mean then an interested reader can check your claims of significance.
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