Standard error
A standard error measures how much a sample statistic would vary across repeated samples.
What a standard error is
A standard error describes the typical sampling variation of an estimate. If a study were repeated many times with the same design, each sample could produce a slightly different mean, proportion, regression coefficient, or other statistic. The standard error measures how spread out those repeated-sample estimates would be.
Sampling distributions
The idea depends on a sampling distribution: the distribution of a statistic across repeated samples. For the sample mean, the standard error is the standard deviation of the distribution of sample means. It is about uncertainty in the estimate, not the spread of individual observations.
Standard error of the mean
For independent observations, the standard error of the mean is commonly estimated as s divided by the square root of n, where s is the sample standard deviation and n is the sample size. Larger samples usually reduce the standard error because sample means cluster more tightly around the population mean.
Standard error versus standard deviation
Standard deviation describes variability among data values. Standard error describes uncertainty in an estimate calculated from those values. A dataset can have a large standard deviation but a smaller standard error if the sample size is large enough.
Confidence intervals
Confidence intervals often combine an estimate, a critical value, and a standard error. A rough 95 percent interval for a mean in a large simple sample is the estimate plus or minus about 1.96 standard errors, though the exact method depends on the design, sample size, and model.
Beyond means
Standard errors are used for many estimates, including proportions, differences, odds ratios, regression coefficients, and survey estimates. In complex surveys or regression models, the right standard error may require weights, clustering, robust methods, bootstrap methods, or design-based formulas.
Limits
A small standard error does not fix bias, bad measurement, nonrandom sampling, confounding, or model misspecification. It only speaks to sampling variability under the method used. Precision is useful, but a precise biased estimate can still be wrong.
Why it matters
Standard error turns a single estimate into an uncertainty-aware result. It helps readers judge whether a sample estimate is precise enough to support a conclusion, compare estimates, or plan whether more data are needed.