Reference definitions and notation for mathematical functions.
Math
Standard deviation
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Your result
Mean5
Variance4
Formulaσ = √[Σ(x − mean)² ÷ N] (or n−1 for a sample)
- Calculate the mean.
- Square every deviation.
- Average the squared deviations and take the square root.
Worked example: Population standard deviation of 2,4,4,4,5,5,7,9 is 2.
Accuracy notice: The arithmetic follows the displayed formula. Its practical accuracy still depends on valid inputs, appropriate units, and display rounding.
Understand the result before using it
Standard deviation uses values and data type to calculate its result. It applies a defined mathematical relationship to the values you enter and reports the resulting quantity.
Formula, assumptions, and example
Formula
σ = √[Σ(x − mean)² ÷ N] (or n−1 for a sample)
Assumptions
- The entered values and data type are complete, valid, and use the units or format shown beside each field.
- Inputs represent ordinary real numbers unless the tool explicitly accepts another format.
- The displayed operation and order of operations match the formula shown on the page.
Worked example
Population standard deviation of 2,4,4,4,5,5,7,9 is 2.
When to use it
Good uses
- Check homework or a hand calculation.
- Explore how changing an input changes the result.
- Create a quick numerical reference without setting up a spreadsheet.
Limitations
Common mistakes to avoid
- An undefined operation, such as division by zero, cannot produce a valid finite result.
- Rounded display values can differ slightly from the full internal result.
- Check the formula, source data, units, and displayed precision before relying on the result.
Sources and further reading
These references explain the underlying standards or provide authoritative context. Your own contract, institution, clinician, product documentation, local code, or governing standard may be the controlling source.
Measurement and numerical-reference guidance from NIST.
Questions about this calculation
What does the Standard deviation calculate?
Standard deviation uses values and data type to calculate its result. It applies a defined mathematical relationship to the values you enter and reports the resulting quantity. The formula used is: σ = √[Σ(x − mean)² ÷ N] (or n−1 for a sample)
Which inputs affect the Standard deviation result?
The result uses values and data type. Changing any of these values can change the output, so enter them using the labels and units shown.
Is the Standard deviation result exact?
The arithmetic follows the displayed formula. Its practical accuracy still depends on valid inputs, appropriate units, and display rounding.
When should I verify a Standard deviation result?
Verify it whenever the source measurement, unit convention, rounding, or consequences make small differences important.
Clear answers, without the guesswork
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