What is the difference between standard deviation and standard error
Standard deviation measures the spread of individual observations in a dataset, while standard error measures the spread of sample‑means around the true population mean; SE is derived from SD and shrinks as sample size grows.
Statistics · Sampling distributions
The standard deviation (SD) measures how individual observations in a single dataset vary around their mean. It is calculated from the data themselves and expressed in the same units as the original measurements. The standard error (SE) describes how the mean of many possible samples would vary around the true population mean; it is a property of the sampling distribution, not of a single sample.
Key Formulas
Because SE is derived from SD, the two are linked by the size of the sample. The formula shows that as the number of observations grows, the SE shrinks, reflecting increased confidence in the sample mean. In contrast, the SD does not change with ; it remains a characteristic of the underlying variability of the data.
Key ways the two measures differ:
- SD describes variability of individual data points; SE describes variability of the sample mean.
- SD is independent of sample size; SE decreases with larger .
- SD is reported in raw units; SE is used for inference and confidence intervals.
Suppose a sample of 25 students has a mean test score of 78 and a standard deviation of 12 points. The standard error of the mean is points. This means that if we repeatedly drew samples of 25 students from the same population, the sample means would typically differ from the true population mean by about 2.4 points, not 12 points.
Procedure to compute SE from a known SD:
- 1Calculate the square root of the sample size .
- 2Divide the SD by that square root.
The SD tells you how spread out individual scores are; a large SD indicates that scores are widely scattered. The SE tells you how precisely you have estimated the population mean; a small SE indicates a tight confidence interval around the sample mean. When reporting results, include the SD to describe variability of the data and the SE (or a confidence interval) to convey uncertainty of the estimated mean.
Side‑by‑side comparison of SD and SE:
| Aspect | Value |
|---|---|
| What it measures | Variability of individual data points vs variability of sample mean |
| Units | Same as data vs Same as data (but scaled) |
| Depends on n | No vs Yes (divides by ) |
Check yourself
If a sample of 100 observations has an SD of 15, what is the SE of the mean?
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