StATS: What is a standard error?

The standard error is the estimated standard deviation of a statistic. The formula depends on what statistic you are talking about. For example, the standard error of a sample mean is just the sample standard deviation divided by the square root of the sample size. For more complicated statistics, the standard error is also more complicated.

You can use the standard error in two ways. First, the statistic divided by the standard error gives you a way of testing whether the parameter that your statistic is estimating equals zero. You compare this ratio to a t-distribution. In most contexts, the ratio of the statistic divided by its standard error is called a Wald test. For many simple applications, you can also call this a t-test.

A second application of the standard error is the production of confidence intervals. You can get a crude confidence interval by taking your statistic plus or minus two standard errors. A more precise confidence interval would use percentiles from the t-distribution.

There are a lot of subtleties in the use of the standard error, especially in more complex problems. Sometimes, for example, the standard error applies not to the statistic itself, but to the logarithm of that statistic. For example, a logistic regression model will compute an odds ratio for your data, but the standard error refers not to the odds ratio, but to the log odds ratio. In this situation, you need to compute the confidence interval on the log scale and then transform the results back to the original scale of measurement.

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Definitions, Category: Descriptive statistics.