Standard error

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For the computer programming concept, see standard error stream.

The standard error (SE) is the standard deviation of the sampling distribution of a statistic.[1] The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.



Standard error of the mean[edit]


\text{SE}_\bar{x}\ = \frac{s}{\sqrt{n}}




\text{SD}_\bar{x}\ = \frac{\sigma}{\sqrt{n}}


σ is the standard deviation of the population.


  • If  X_1, X_2 , \ldots, X_n are n independent observations from a population that has a mean  \mu and standard deviation  \sigma , then the variance of the total  T = (X_1 + X_2 + \cdots + X_n) is  n\sigma^2.
  • The variance of  T/n must be  \frac{1}{n^2}n\sigma^2=\frac{\sigma^2}{n}.
  • And the standard deviation of  T/n must be  \sigma/{\sqrt{n}} .
  • Of course,  T/n is the sample mean  \bar{x} .



Student approximation when σ value is unknown[edit]

In practical applications, the true value of the σ value is unknown. As a result, we need to use an approximation of the Gaussian sample distribution. In this application the standard error is the standard deviation of a t-student distribution. T-distributions are slightly different from the gaussian, and vary depending on the size of the sample. To estimate the Standard error of a t-student distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals.

Note: The student's probability distribution is a good approximation of the Gaussian when the sample size is over 100.

Assumptions and usage[edit]

If the data are assumed to be normally distributed, quantiles of the normal distribution and the sample mean and standard error can be used to calculate approximate confidence intervals for the mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where \bar{x} is equal to the sample mean, SE is equal to the standard error for the sample mean, and 1.96 is the 0.975 quantile of the normal distribution:

Upper 95% limit = \bar{x} + (\text{SE}\times 1.96) , and
Lower 95% limit = \bar{x} - (\text{SE}\times 1.96) .



Standard error of mean versus standard deviation[edit]


Put simply, the standard error of the sample is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to the population standard deviation as the sample size increases.

Correction for finite population[edit]


When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the error must be corrected by multiplying by a "finite population correction"[7]

    \text{FPC} = \sqrt{\frac{N-n}{N-1}}


Correction for correlation in the sample[edit]




where the sample bias coefficient ρ is the widely used Prais-Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover this formula works for positive and negative ρ alike.[8] See also unbiased estimation of standard deviation for more discussion.

Relative standard error[edit]

The relative standard error of a sample mean is simply the standard error divided by the mean and expressed as a percentage. The relative standard error only makes sense if the variable for which it is calculated cannot have a mean of zero.


See also[edit]


  5. ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". American Statistician (American Statistical Association) 25 (4): 30–32. doi:10.2307/2682923. JSTOR 2682923. 
  7. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample". Journal of the Royal Statistical Society (Blackwell Publishing) 81 (1): 75–81. doi:10.2307/2340569. JSTOR 2340569.  (Equation 1)
  9. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression". Statistical Notes (Hyattsville, MD: U.S. National Center for Health Statistics) (24). Retrieved 17 July 2014.