Standard error

Standard error is a central concept of statistics. It is the standard deviation of the sampling distribution associated with the estimation method. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample.

For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard errors above and below the actual value.

For example, the sample mean is the usual estimator of a population mean. However, different samples drawn from that same population would in general have different values of the sample mean. The standard error of the mean is the standard deviation of those sample means over all samples drawn from the population. Secondarily, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analysed at the time.

The term standard error is derived from the fact that, as long as the estimator is unbiased, the standard deviation of the error (the difference between the estimate and the true value) is the same as the standard deviation of the estimates themselves; this is true since the standard deviation of the difference between the random variable and its expected value is equal to the standard deviation of a random variable itself.

In many practical applications, the true value of the standard deviation is usually unknown. As a result, the term standard error is often used to refer to an estimate of this unknown quantity. In such cases it is important to be clear about what has been done and to attempt to take proper account of the fact that the standard error is only an estimate.

Unfortunately, often this is not possible, and it may then be better to use an approach that avoids using a standard error. In some cases, the standard error may usefully be used to provide an indication of the size of the uncertainty.

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