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Standard Deviation and Standard Error from Data — result sheet
Calculate a sample standard deviation, standard error of the mean, and relative standard error from raw observations.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: s = √[Σ(x−x̄)²/(n−1)]; standard error = s/√n; relative standard error = standard error/|x̄| when x̄ ≠ 0.
Standard deviation describes observation-to-observation spread, while standard error describes the modeled uncertainty of a sample mean under a simple independent-sample interpretation. Reporting both prevents the two measures from being conflated.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Sample values
Worked example
| Input | Value |
|---|---|
| Sample values | 4, 7, 9, 10, 15 |
Sample standard deviation is about 4.0620; standard error is about 1.8166; relative standard error is about 20.184%.
Assumptions and limits
- The entered values are treated as a sample of independent observations with a meaningful arithmetic mean.
- The sample divisor n−1 is used for the standard deviation and n must be at least two.
- Standard error is a model-based summary; it does not capture bias, dependence, nonresponse, or a nonrepresentative sample.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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