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Standard error

For the computer programming concept, see standard error stream.
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 deviations above and below the actual value.

The standard error (SE)[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[1] The standard error is a key ingredient in producing confidence intervals.[3]

The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size.[1] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).

  1. ^ a b c Altman, Douglas G; Bland, J Martin (2005-10-15). "Standard deviations and standard errors". BMJ: British Medical Journal. 331 (7521): 903. doi:10.1136/bmj.331.7521.903. ISSN 0959-8138. PMC 1255808. PMID 16223828.
  2. ^ Everitt, B. S. (2003). The Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 978-0-521-81099-9.
  3. ^ Wooldridge, Jeffrey M. (2023). "What is a standard error? (And how should we compute it?)". Journal of Econometrics. 237 (2, Part A). doi:10.1016/j.jeconom.2023.105517. ISSN 0304-4076.

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