Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean. It quantifies the precision of a sample statistic as an estimate of the corresponding population parameter. A smaller standard error indicates a more precise estimate. The standard error of the mean is calculated as the sample standard deviation divided by the square root of the sample size.

The standard error is the building block for constructing confidence intervals and conducting hypothesis tests. It translates the variability observed in a sample into a measure of uncertainty about the population parameter. Researchers use the standard error to determine how much the sample mean might differ from the true population mean, to construct confidence intervals that express the range of plausible values for the parameter, and to calculate test statistics (such as the t statistic) for hypothesis testing. Ignoring the standard error leaves readers unable to tell how far a sample estimate could plausibly be from the population value, so chance findings can be overinterpreted; the standard error describes random error only, not systematic error (bias).

For the sample mean, the standard error equals s divided by the square root of n, where s is the sample standard deviation and n is the sample size. This formula reveals that the standard error decreases as sample size increases, reflecting the greater precision of larger samples. Quadrupling the sample size halves the standard error.

The standard error can be computed for any sample statistic, not just the mean. The standard error of a proportion, the standard error of a regression coefficient, and the standard error of a correlation coefficient each have specific formulas. In all cases, the standard error represents the expected variability of the statistic across repeated samples.

Confidence intervals are constructed using the standard error. For example, the 95% confidence interval for the mean is approximately the sample mean plus or minus 1.96 times the standard error (using the normal approximation, or the appropriate t critical value for small samples). The width of the confidence interval is directly proportional to the standard error.

In a fixed effect meta analysis, each study's weight is the inverse of its squared standard error; in a random effects meta analysis, the between study variance is added to the squared standard error first, so the weights depend on heterogeneity as well. Studies with smaller standard errors (more precise estimates) receive more weight in the pooled analysis. The standard error of the pooled estimate reflects the combined precision of all included studies.

Report standard errors when presenting the precision of sample statistics, constructing confidence intervals, or providing information needed for meta analysis. Use the standard error when the goal is to characterize the uncertainty of an estimate about the population parameter. Use the standard deviation when the goal is to describe the variability of individual observations in the sample. Always clarify which measure is being reported, as confusion between the two is common and consequential.

A common mistake is confusing the standard error with the standard deviation. Reporting the standard error in place of the standard deviation makes the data appear less variable than they actually are (since SE is always smaller than SD for samples larger than one). Another error is using the standard error to construct error bars on graphs without specifying what the error bars represent, leaving readers unable to interpret the display. Researchers sometimes calculate the standard error using the wrong denominator (using n instead of n minus 1 in the standard deviation calculation) or apply the normal distribution critical value when the t distribution is more appropriate for small samples.

A study measures serum vitamin D levels in 100 elderly adults, finding a sample mean of 22 ng/mL with a standard deviation of 8 ng/mL. The standard error of the mean is 8 divided by the square root of 100, which equals 0.8 ng/mL. The 95% confidence interval for the population mean is 22 plus or minus 1.96 times 0.8, or approximately 20.4 to 23.6 ng/mL. If the study had included 400 participants with the same standard deviation, the standard error would be 0.4 ng/mL and the confidence interval would narrow to 21.2 to 22.8 ng/mL.