Keep the target of variation visible
Imagine a fictional sample of 100 independent observations with a standard deviation of 20 units. Under the usual simple-sample assumptions, the estimated standard error of the mean is 20 divided by the square root of 100, or 2 units.
The 20 describes spread among observations. The 2 describes uncertainty in the estimated mean due to sampling. It does not mean most observations fall within two units of the mean.
What happens when the sample grows?
If a comparable sample contains 400 independent observations with the same standard deviation, the corresponding standard error is 1 unit. The mean can be estimated more precisely even though individual variation remains 20 units.
The calculation depends on its assumptions. Clustering, weighting, dependence, or a different estimator can require a different standard-error calculation.
The error bar needs a label
| Label | Main interpretation |
|---|---|
| SD | Spread of observations |
| SE | Sampling variability of the estimate |
| Confidence interval | An interval constructed to express uncertainty under stated methods |
The NIST glossary defines standard error in relation to a statistic's sampling distribution. That distinction is easy to lose when a graph labels all three possibilities merely “error bars.”
Why it matters to a reader
A graph with small standard-error bars can make two groups look internally uniform even when participants vary widely. If the question concerns individual experiences, a distribution plot or standard deviation may be more informative.
If the question concerns the precision of a mean difference, its standard error or confidence interval may be appropriate. The choice is not between a truthful and an untruthful statistic. It is between summaries of different kinds of variation. Reading the label prevents one from answering the other's question.
