# Standard error (statistics)

In statistics, the standard error of a measurement, value or quantity is the standard deviation of the process by which it was generated.

Standard errors provide simple measures of uncertainty in a value and are often used because:

• If the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated in many cases;
• Where the probability distribution of the value is known, they can be used to calculate an exact confidence interval; and
• Where the probability distribution is unknown, relationships like Chebyshev's or the Vysochanskiï-Petunin inequality can be used to calculate a conservative confidence interval.

The standard error of a sample from a population is the standard deviation of the sampling distribution and may estimated by the formula:

[itex]\frac{\sigma}{\sqrt{N}}[itex]

where [itex]\sigma[itex] is the standard deviation of the population distribution and N is the size (number of items) in the sample.

A very important implication of this formula is that you must quadruple the sample size (4X) to achieve half (1/2) the measurement error. When designing statistical studies where cost is a factor, this may have a factor in understanding cost-benefit tradeoffs.

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