How is the standard error different from the standard deviation of the data?

Ask what is varying. If you mean the differences among individual scores in one group, you mean standard deviation. If you mean how much a statistic—often the group’s mean—would change from sample to sample, you mean standard error. Standard deviation describes the spread of individual data values around their mean; standard error describes sampling variability in a statistic. [S1]

A quick comparison:

1. Standard deviation: variation among the observations you have.
2. Standard error: variation in a statistic across repeated samples, or an estimate of that sampling variability.

They are related, but they answer different questions. A standard error is not another name for the spread of individual scores.

Suppose, purely as a made-up example, a teacher is studying quiz scores from a class. The standard deviation would describe how much individual students’ scores differ from the class mean. It addresses: “How varied were the students’ scores?”

The standard error of the mean addresses a different question: “How much might the mean differ if we repeatedly took samples of this size from the population?” For the mean, when the population standard deviation σ is known, the standard error is σ/√n. [S1]

For a worked numerical illustration, assume the population standard deviation is 10 points and the sample size is 25. Then the standard error of the mean is 10/√25 = 2 points. This does not say students’ scores are typically only 2 points apart. The assumed standard deviation describes score spread; the calculated standard error describes sampling variability in the mean. These are illustrative assumptions, not results from a real class.

Not necessarily. A smaller standard error means less sampling variability in the statistic under the conditions of the calculation; it does not by itself show that individual scores were closer together. In the known-population-standard-deviation example, increasing n makes σ/√n smaller, even though that formula alone tells you nothing about whether the students’ scores are similar. [S1]

Keep the two interpretations separate:

1. Use standard deviation to describe how spread out the observed scores are.
2. Use standard error when discussing the sample mean’s sampling variability.

Also check what the reported standard error belongs to. It might describe a mean or another statistic; do not assume it is always the standard error of a mean.

Describe only what the report establishes. You could write: “The report gives a standard error for the sample mean, which concerns the mean’s sampling variability; it is not a measure of the spread of individual quiz scores.” If the report does not say how the value was calculated or what assumptions were used, don’t fill in those details yourself. Ask the instructor or check the report’s methods.

The formula σ/√n applies to the standard error of the mean when the population standard deviation is known. The cited explanation distinguishes that formula from a method that estimates the standard error using a sample standard deviation; it does not establish which calculation your report used. [S1]

Before writing, identify the statistic, its standard error, the sample size if given, and whether the calculation method is stated. Report the standard deviation separately if your goal is to describe variation among students’ scores.