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MasterMath

Standard Error Calculator

The standard error of the mean says how much the mean would wobble if you repeated the sampling. It is not the same as the standard deviation, and confusing the two is the most repeated mistake in applied statistics.

Standard error of the mean

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Standard error of the mean—
Sample mean—
Standard deviation—
Number of values—
95 % interval—
95 % margin—
MeasureValue

How this was worked out

    The formula

    SE = s / √n

    What it means

    Standard deviation measures how far the values spread from each other; standard error measures how far the mean would move from one sample to the next. They are different things: the first is a property of the data and does not fall by collecting more, the second is a property of your estimate and falls with the square root of n. So quadrupling the sample halves the standard error rather than quartering it, and there is the law of diminishing returns behind every survey.

    How to work it out by hand

    1. Work out the sample standard deviation of the values
    2. Count how many values there are
    3. Take the square root of that count
    4. Divide the deviation by that square root

    What is worth knowing

    An error bar on a chart may be a standard deviation or a standard error, and they are very different: the standard error is always smaller and makes the data look more precise than it is. A serious paper states which one is drawn. If it does not say, there is no way to know what you are looking at, and that alone is a good reason to distrust the chart.

    Frequently asked questions

    What is the difference from standard deviation?

    Deviation measures how much the values vary; standard error, how much the mean would vary if you repeated the study. The second depends on sample size and the first does not.

    Why divide by the square root of n?

    Because the variance of the mean is the variance of the data over n, and the standard error is its square root. That is where the root comes from.

    How much data do I need to halve it?

    Four times as much. The error falls with the square root, so improvements get steadily more expensive.

    Can I use it with small samples?

    It is worked out the same way, but with fewer than thirty values a confidence interval should use Student's t rather than the normal.

    Which error bar should I draw?

    Whichever answers your question: the deviation if you are talking about the spread of the data, the standard error if you are talking about the precision of the mean. And say so in the caption.