The standard error of a method of measurement or estimate is the estimated standard deviation of the error in that method. Namely, it is the standard deviation of the difference between the measured or estimated values and the true values. Notice that the true value is, by definition, unknown and this implies that the standard error of an estimate is itself an estimated value. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... The word error has different meanings in different domains. ...

In particular, for instance, the standard error of a sample statistic (such as sample mean) is the estimated standard deviation of the error in the process by which it was generated. In other words, it is the standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or S_E. A sample statistic is the result of some mathematical process applied to a sample used to infer a population parameter. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... Process (lat. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ...

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
• As the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.

The standard error of the mean of a sample from a population is the standard deviation of the sampling distribution of the mean, and may be estimated by the formula: Partial plot of a function f. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In statistics, a confidence interval (CI) for a population parameter is an interval between two numbers with an associated probability p which is generated from a random sample of an underlying population, such that if the sampling was repeated numerous times and the confidence interval recalculated from each sample according... In probability theory, Chebyshevs inequality (also known as Tchebysheffs inequality, Chebyshevs theorem, or the Bienaymé-Chebyshev inequality), named after Pafnuty Chebyshev, who first proved it, states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a... In probability theory, the Vysochanskiï-Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variables mean. ... Sample size, usually designated N, is the number of repeated measurements in a statistical sample. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ...

where

is an estimate of the standard deviation σ of the population, and

n is the size (number of items) of the sample.

See also

• sampling distribution
• standard deviation