Standard error

The standard error or sampling error is a measure of dispersion for an estimator ${\hat {\vartheta }}$ for an unknown parameter $\vartheta$ the population. The standard error is defined as the standard deviation σ $\sigma ({\hat {\vartheta }})=+{\sqrt {\operatorname {Var} ({\hat {\vartheta }})}}$ of the estimator, ${\hat {\vartheta }}$ , that is, the positive square root of the variance. In the natural sciences and metrology, the term standard uncertainty, coined by the GUM, is also used.

Therefore, for an expectation-true estimator, the standard error is a measure of the average deviation of the estimated parameter value from the true parameter value. The smaller the standard error, the more accurately the unknown parameter can be estimated using the estimator. The standard error depends, among other things, on

the sample size and
of the variance in the population.

In general, the larger the sample size, the smaller the standard error; the smaller the variance, the smaller the standard error.

The standard error also plays an important role in the calculation of estimation errors, confidence intervals and test statistics.

Interpretation

The standard error provides information about the quality of the estimated parameter. The more individual values there are, the smaller the standard error, and the more accurately the unknown parameter can be estimated. The standard error makes the measured dispersion (standard deviation) of two data sets with different sample sizes comparable by normalizing the standard deviation to the sample size.

If multiple samples are used to estimate the unknown parameter, the results will vary from sample to sample. Of course, this variation does not come from a variation in the unknown parameter (because that is fixed), but from random influences, such as measurement inaccuracies. The standard error is the standard deviation of the estimated parameter in many samples. In general, halving the standard error requires quadrupling the sample size.

In contrast, the standard deviation represents the dispersion actually present in a population, which is also present with the highest measurement accuracy and an infinite number of individual measurements (e.g. weight distribution, height distribution, monthly income). It shows whether the individual values are close together or whether there is a wide spread in the data.

Example

Suppose one examines the population of children attending grammar schools in terms of their intelligence performance. The unknown parameter is therefore the mean intelligence performance of the children attending a grammar school. Now, if at random from this population a sample of size (i.e. with children) is drawn at random from this population, then the mean value can be calculated from all measurement results. Now, if another randomly drawn sample with the same number of children is drawn after this sample and its mean is calculated, the two means will not match exactly. If one draws a large number of other random samples of size then the dispersion of all empirically determined mean values around the mean value of the population can be determined. This dispersion is the standard error. Since the mean of the sample means is the best estimator of the population mean, the standard error is the dispersion of the empirical means around the population mean. It does not represent the dispersion of children's intelligence, but the precision of the calculated mean.

Questions and Answers

Q: What is the standard error?

A: The standard error is the standard deviation of the sampling distribution of a statistic.

Q: Can the term standard error be used for an estimate of the standard deviation?

A: Yes, the term standard error can be used for an estimate (good guess) of that standard deviation taken from a sample of the whole group.

Q: How does one estimate the average for a whole group?

A: The average of some part of a group (called a sample) is the usual way to estimate the average for the whole group.

Q: Why is it difficult to measure the whole group?

A: It is often too hard or too costly to measure the whole group.

Q: What is the standard error of the mean, and what does it determine?

A: The standard error of the mean is a way to know how close the average of the sample is to the average of the whole group. It is a way of knowing how sure one can be about the average from the sample.

Q: Is the true value of the standard deviation of the mean usually known in real measurements?

A: No, the true value of the standard deviation of the mean for the whole group is usually not known in real measurements.

Q: How does the number of measurements in a sample affect the accuracy of the estimate?

A: The more measurements there are in a sample, the closer the guess will be to the true number for the whole group.