They answer different questions
Standard deviation describes the data: how spread out the individual values are. Standard error of the mean describes the estimate: how much the mean itself would move if you repeated the experiment. One is a property of the thing you measured; the other is a property of how well you measured it.
The formulas
SD = √(Σ(x − mean)² ÷ (n − 1))
SEM = SD ÷ √n
Ten values with a mean of 20 and an SD of 4 give an SEM of 4 ÷ √10 = 1.26. With 100 values and the same SD of 4, the SEM falls to 0.4.
Why SEM always looks better
SD does not shrink as you collect more data; it settles on the real spread of the population. SEM shrinks with √n without limit, so plotting SEM makes error bars smaller purely by adding samples. That is a correct description of a more precise mean, and a misleading picture of the variability, which is why figures that do not say which one they used are unreadable.
Which to report
- SD: when you are describing a sample. Patient ages, cell diameters, the range of values a reader should expect to see.
- SEM or a confidence interval: when you are comparing means, because that is the uncertainty the comparison rests on.
- 95% confidence interval: usually clearer than SEM, and roughly mean ± 1.96 × SEM for a large sample. It states a range rather than a quantity the reader has to convert.
Whatever you choose, say so in the caption, together with n. "Mean ± SD, n = 12" takes four words and removes all ambiguity.
Reading error bars
Overlapping SEM bars do not prove there is no difference, and non-overlapping ones do not prove there is. For two groups, bars that are more than about one SEM apart are often significant at the 5% level, but the only honest answer comes from the test and the P value, not from looking at the picture.
Sample or population
The n − 1 in the SD formula is Bessel's correction: dividing by n underestimates the spread when you only have a sample. Use the n version only when your numbers are every member of the group, which in practice almost never happens in a lab.
Tools
The Standard Deviation Calculator gives SD, SEM, variance and the quartiles from a pasted list, for a sample or a population. The t-Test Calculator compares two sets of numbers and gives the P value, and the Weighted Average Calculator handles means where the values do not count equally.