standard error of the mean
/ STAN-derd ERR-or uv thuh meen /
Suppose you weigh a coin three times and average the results, then do the whole thing again, and again. Each batch of three gives a slightly different average. The standard error of the mean tells you how much those batch-averages themselves would jump around — not how scattered single weighings are, but how shaky the average of a few is.
The standard error of the mean is the standard deviation divided by the square root of the number of measurements. It measures the uncertainty in the average, and it shrinks as you take more readings. Crucially, because of that square root, cutting the uncertainty in half requires four times as many measurements, so improvement gets steadily more expensive.
It matters because the average, not the single reading, is what you usually report, so its uncertainty is what belongs on the result — it is the standard error, not the standard deviation, that goes into a confidence interval for the mean. The common confusion is mixing the two up: the standard deviation describes the spread of the data, while the standard error describes the precision of their average.
Nine measurements have a standard deviation of 0.6 percent. The standard error of the mean is 0.6 divided by the square root of nine, that is 0.6 over 3, giving 0.2 percent for the uncertainty in their average.
The average of nine readings is three times surer than one alone.
Because the standard error keeps falling with more measurements while the standard deviation does not, error bars on a graph should always state which of the two they represent — they tell very different stories.