The difference between the SD and SEM

It is easy to be confused about the difference between the standard deviation (SD) and the standard error of the mean (SEM). 

The SD quantifies scatter — how much the values vary from one another. 

The SEM quantifies how accurately you know the true mean of the population. The SEM gets smaller as your samples get larger. This makes sense, because the mean of a large sample is likely to be closer to the true population mean than is the mean of a small sample. 

The SD does not change predictably as you acquire more data. The SD quantifies the scatter of the data, and increasing the size of the sample does not change the scatter. The SD might go up, or it might go down; you can't predict. On average, the SD will stay the same as sample size gets larger. 

If the scatter is caused by biological variability, you probably will want to show the variation. In this case, report the SD rather than the SEM. If you are using an in vitro system with no biological variability, the scatter can only result from experimental imprecision. In this case, you may not want to show the scatter, but instead show how well you have assessed the mean. Report the mean and SEM, or the mean with 95% confidence interval. 

You should choose to show the SD or SEM based on the source of the variability and the point of the experiment. In fact, many scientists choose the SEM simply because it is smaller so creates shorter error bars.