Statistics & data
Standard Deviation & Standard Error Calculator
Paste repeated measurements separated by commas, spaces or new lines. Get both measures of spread and the standard error of the mean.
Your measurements
Start with the example, or enter your own values. Use a dot for decimals and e for scientific notation.
Calculated locally. No account or uploads.
Result & working
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Step-by-step working
The formula
x̄ = Σxᵢ/n
σ = √[Σ(xᵢ − x̄)²/n]
s = √[Σ(xᵢ − x̄)²/(n − 1)]
SEM = s/√n
| Symbol | Meaning |
|---|---|
| n | Number of measurements. |
| x̄ | Arithmetic mean. |
| σ | Population standard deviation: denominator n. |
| s | Sample standard deviation: denominator n − 1. |
| SEM | Estimated standard error of the mean using sample SD. |
When to use this calculator
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.
Worked example
The calculator opens with these example values. All steps below are available even with JavaScript disabled.
- 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
Mean = 5
Common mistakes
- 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.