What it means
Standard deviation describes spread in the data itself. Standard error describes the spread of an estimate made from the data, so the two are linked but answer different questions.
For the sample mean, NIST gives the formula as the standard deviation s divided by the square root of N, where N is the number of observations. OpenStax calls the standard deviation of the sample mean the standard error of the mean and links it to the central limit theorem.
As sample size grows, the means cluster more tightly around the true mean. The square root matters, because to cut the standard error in half you need four times the data.
Doubling the sample only reduces it by about 29%, since 1 divided by the square root of 2 is about 0.71. Standard error is the building block of confidence intervals and tests.
OpenStax gives a critical value of 1.96 for a 95% confidence interval based on the normal distribution, so a rough 95% range for a mean is the sample mean plus or minus 1.96 standard errors. A related use is the margin of error in polls and surveys, which is built from a standard error times a critical value, and a reader who sees only the margin should ask for the sample size because a small sample gives a wide margin whatever the headline says.
In finance, it appears in return estimates, regression coefficients and survey results. An average monthly return from 36 months of data has a standard error that may be large, so the estimate can mislead, which is why a fund's past average is a weak guide on its own.
A fund manager's track record, a regression of sales on advertising spend and an employee survey all carry a standard error that tells the reader how much to trust the headline number. There are limits.
The simple formula assumes a random sample and independent observations, so data that is clustered, trending or autocorrelated, such as stock returns in a volatile year, may need other methods.
In practice
Real-world examples.
Example
A fictional fund has a monthly return standard deviation of 12% across 36 months. The standard error of its average return is 12 / sqrt(36) = 2.0%. If the average was 1.5% a month, the 95% range of about plus or minus 3.92% runs from -2.42% to 5.42% and includes zero.
Example
A fictional shop surveys customers on spending. With 144 people and a sample standard deviation of 12, the standard error is 12 / sqrt(144) = 1.0. If the average is 40, the rough 95% range is 38.04 to 41.96.
Example
A fictional analyst wants a standard error of 1.0 for an index return with standard deviation 15. The needed sample is 225, since 15 / sqrt(225) = 1.0. Cutting it to 0.5 requires 900 observations, four times as many.
Formula
Calculation
Standard error of the mean = s / sqrt(N). With s = 12 and N = 36 this is 12 / 6 = 2.0.
Approximate 95% range = sample mean +/- 1.96 x standard error. With a mean of 1.5 and SE of 2.0, the range is 1.5 +/- 3.92, or -2.42 to 5.42.
Effect of sample size, holding s = 12 constant:
- N = 9 gives SE = 12 / 3 = 4.0.
- N = 36 gives SE = 12 / 6 = 2.0.
- N = 144 gives SE = 12 / 12 = 1.0.
Each fourfold increase in observations halves the standard error, which is why precision becomes expensive quickly.Case study
Seen in the real world.
This case study is fictional and illustrative. Hana, an analyst, compares two funds using 36 months of returns. Fund A averages 1.2% a month and Fund B averages 0.9%, and both have a monthly standard deviation of 6%. Each fund's standard error is 6 / sqrt(36) = 1.0%. The gap between the two averages is 0.3%, which is small beside a standard error of 1.0% each.
The data cannot separate them. She asks for 144 months instead. The standard error falls to 6 / 12 = 0.5%, and the picture is clearer. She tells her team that more data narrows the range, but a bigger sample never repairs a biased one.
Watch out
Common mistakes.
- Mixing up standard error with standard deviation, when one describes data spread and the other describes estimate precision.
- Expecting a larger sample to help linearly, when the square root means returns diminish.
- Applying the simple formula to biased or dependent data, when the sample must be random and independent.
Questions
People also ask.
What is standard error?
It is the standard deviation of a sample statistic, such as a mean, across repeated samples. It shows how precise an estimate is. A smaller value means a tighter estimate.
How do you calculate it for a mean?
Divide the sample standard deviation by the square root of the number of observations. NIST gives this as s divided by the square root of N. With s = 12 and N = 36 the result is 2.0.
How is it different from standard deviation?
Standard deviation describes spread in the data. Standard error describes spread in an estimate. A big standard deviation can still give a small standard error with a large sample.
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