Statistics & data
Standard deviation & error
Find sample SD, population SD and standard error.
Your measurements
Number formats & units
Use decimal dots in datasets. Single values also accept decimal commas and scientific notation such as 1.2e-3 or 1.2 × 10⁻³.
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Result & working
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Interpretation & assumptions
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Formula & example
The relationship
What the variables mean
| Symbol | Meaning |
|---|---|
| Number of measurements. | |
| Arithmetic mean. | |
| Population standard deviation: denominator n. | |
| Sample standard deviation: denominator n − 1. | |
| Estimated standard error of the mean using sample SD. |
Worked example
- Measurements
- 2, 4, 4, 4, 5, 5, 7, 9
Mean = 5
Follow the example steps
- n = 8
- x̄ = Σxᵢ / n = 40 / 8 = 5
- Σ(xᵢ − x̄)² = 32
- σ = √[Σ(xᵢ − x̄)² / n] = √(32 / 8) = 2
- s = √[Σ(xᵢ − x̄)² / (n − 1)] = √(32 / 7) = 2.13809
- SEM = s / √n = 2.13809 / √8 = 0.7559289
Common mistakes
Standard deviation and standard error describe different things.
Incorrect approach
For 2, 4, 6: sample SD = 2/√3
Correct approach
Sample SD = 2; SEM = 2/√3 ≈ 1.155
More checks before you submit
- Do not interchange SD and SEM in error bars without explaining what the bars represent.
- Do not divide the population SD by √n and label it the sample-based SEM.
- Do not silently delete an outlier because it changes the answer.
Method notes
Use population SD to describe the complete set of values you are treating as the population. Use sample SD to estimate variability in a wider population from a sample.
For independent repeated measurements, SEM estimates the sampling variability of their mean. SD describes the spread of individual observations. Repetition can reduce SEM, but it does not remove a systematic calibration error. One value has zero population spread; sample SD and SEM need at least two values.