Standard error

Suppose a statistically independent sample of ${\displaystyle n}$ observations ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ is taken from a statistical population with a standard deviation of ${\displaystyle \sigma }$. The mean value calculated from the sample, ${\displaystyle {\bar {x}}}$, will have an associated standard error on the mean, ${\displaystyle {\sigma }_{\bar {x}}}$, given by:[1]

$${\displaystyle {\sigma }_{\bar {x}}={\frac {\sigma }{\sqrt {n}}}.}$$

Practically this tells us that when trying to estimate the value of a population mean, due to the factor ${\displaystyle 1/{\sqrt {n}}}$, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations.

The standard deviation ${\displaystyle \sigma }$ of the population being sampled is seldom known. Therefore, the standard error of the mean is usually estimated by replacing ${\displaystyle \sigma }$ with the sample standard deviation ${\displaystyle \sigma _{x}}$ instead:

$${\displaystyle {\sigma }_{\bar {x}}\ \approx {\frac {\sigma _{x}}{\sqrt {n}}}.}$$

As this is only an estimator for the true "standard error", it is common to see other notations here such as:

$${\displaystyle {\widehat {\sigma }}_{\bar {x}}:={\frac {\sigma _{x}}{\sqrt {n}}}\qquad {\text{ or }}\qquad {s}_{\bar {x}}\$$ :={\frac {s}{\sqrt {n}}}.}

A common source of confusion occurs when failing to distinguish clearly between:

• the standard deviation of the population (${\displaystyle \sigma }$),
• the standard deviation of the sample (${\displaystyle \sigma _{x}}$),
• the standard deviation of the mean itself (${\displaystyle \sigma _{\bar {x}}}$, which is the standard error), and
• the estimator of the standard deviation of the mean (${\displaystyle {\widehat {\sigma }}_{\bar {x}}}$, which is the most often calculated quantity, and is also often colloquially called the standard error).

When the sample size is small, using the standard deviation of the sample instead of the true standard deviation of the population will tend to systematically underestimate the population standard deviation, and therefore also the standard error. With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect.[4] Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20.[5] See unbiased estimation of standard deviation for further discussion.

The standard error on the mean may be derived from the variance of a sum of independent random variables,[6] given the definition of variance and some simple properties thereof. If ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ is a sample of ${\displaystyle n}$ independent observations from a population with mean ${\displaystyle {\bar {x}}}$ and standard deviation ${\displaystyle \sigma }$, then we can define the total

$${\displaystyle T=(x_{1}+x_{2}+\cdots +x_{n})}$$

which due to the Bienaymé formula, will have variance

$${\displaystyle \operatorname {Var} (T)={\big (}\operatorname {Var} (x_{1})+\operatorname {Var} (x_{2})+\cdots +\operatorname {Var} (x_{n}){\big )}=n\sigma ^{2}.}$$

where we've approximated the standard deviations, i.e., the uncertainties, of the measurements themselves with the best value for the standard deviation of the population. The mean of these measurements ${\displaystyle {\bar {x}}}$ is simply given by

$${\displaystyle {\bar {x}}=T/n.}$$

The variance of the mean is then

$${\displaystyle \operatorname {Var} ({\bar {x}})=\operatorname {Var} \left({\frac {T}{n}}\right)={\frac {1}{n^{2}}}\operatorname {Var} (T)={\frac {1}{n^{2}}}n\sigma ^{2}={\frac {\sigma ^{2}}{n}}.}$$

The standard error is, by definition, the standard deviation of ${\displaystyle {\bar {x}}}$ which is simply the square root of the variance:

$${\displaystyle \sigma _{\bar {x}}={\sqrt {\frac {\sigma ^{2}}{n}}}={\frac {\sigma }{\sqrt {n}}}.}$$

For correlated random variables the sample variance needs to be computed according to the Markov chain central limit theorem.

There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size ${\displaystyle N}$ is a random variable whose variation adds to the variation of ${\displaystyle X}$ such that,

$${\displaystyle \operatorname {Var} (T)=\operatorname {E} (N)\operatorname {Var} (X)+\operatorname {Var} (N){\big (}\operatorname {E} (X){\big )}^{2}}$$

[7]

which follows from the law of total variance.

If ${\displaystyle N}$ has a Poisson distribution, then ${\displaystyle \operatorname {E} (N)=\operatorname {Var} (N)}$ with estimator ${\displaystyle n=N}$. Hence the estimator of ${\displaystyle \operatorname {Var} (T)}$ becomes ${\displaystyle nS_{X}^{2}+n{\bar {X}}^{2}}$, leading the following formula for standard error:

$${\displaystyle \operatorname {Standard~Error} ({\bar {X}})={\sqrt {\frac {S_{X}^{2}+{\bar {X}}^{2}}{n}}}}$$

(since the standard deviation is the square root of the variance).

In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[8]

Put simply, the standard error of the sample mean 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.[9] If the population standard deviation is finite, the standard error of the mean 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 approximate the population standard deviation as the sample size increases.

The formula given above for the standard error assumes that the population is infinite. Nonetheless, it is often used for finite populations when people are interested in measuring the process that created the existing finite population (this is called an analytic study). Though the above formula is not exactly correct when the population is finite, the difference between the finite- and infinite-population versions will be small when sampling fraction is small (e.g. a small proportion of a finite population is studied). In this case people often do not correct for the finite population, essentially treating it as an "approximately infinite" population.

If one is interested in measuring an existing finite population that will not change over time, then it is necessary to adjust for the population size (called an enumerative study). When the sampling fraction (often termed f) is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a ''finite population correction'' (a.k.a.: FPC):[10] [11]

$${\displaystyle \operatorname {FPC} ={\sqrt {\frac {N-n}{N-1}}}}$$

which, for large N:

$${\displaystyle \operatorname {FPC} \approx {\sqrt {1-{\frac {n}{N}}}}={\sqrt {1-f}}}$$

to account for the added precision gained by sampling close to a larger percentage of the population. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N.

This happens in survey methodology when sampling without replacement. If sampling with replacement, then FPC does not come into play.

Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. The unbiased standard error plots as the ρ = 0 diagonal line with log-log slope −1⁄2.

If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f:

$${\displaystyle f={\sqrt {\frac {1+\rho }{1-\rho }}},}$$

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.[12] See also unbiased estimation of standard deviation for more discussion.