Calculated using standard mathematical and industry formulas.
Free online Standard Deviation Calculator calculator for academic, engineering, and statistical math.
How to use this calculator?
Enter your data values separated by commas or spaces.
Read both sample SD (s) and population SD (σ), plus variance for each.
Standard error and coefficient of variation are also computed.
Formula & calculation method
Standard deviation formulas
σ² = Σ(xᵢ − μ)² / N · s² = Σ(xᵢ − x̄)² / (N−1) · SE = s / √N
σ = population SD, s = sample SD, μ = population mean, x̄ = sample mean.
Step-by-step calculation example
[2,4,4,4,5,5,7,9]: mean = 5, Σ(xᵢ−μ)² = 32, σ² = 4, σ = 2, s² ≈ 4.57, s ≈ 2.14.
Key insights to remember
Sample SD (s) divides by N−1 (Bessel's correction) to correct for estimation bias when working with a sample.
Population SD (σ) divides by N — use this only when you have the entire population.
For dataset [2,4,4,4,5,5,7,9]: mean = 5, population σ = 2, sample s ≈ 2.14.
Standard error SE = s/√N estimates how far the sample mean is from the population mean.
How to interpret your results?
Sample vs population SD
Use sample SD (s) when your data is a sample from a larger population. Use population SD (σ) only when you have every member of the population.
Coefficient of variation
CV = (s / |mean|) × 100. Expresses dispersion as a percentage of the mean — useful for comparing variability across datasets with different units or scales.
How this differs from the Statistics Calculator
This calculator specialises in variance and standard deviation outputs. The Statistics Calculator adds quartiles, IQR, skewness, and a five-number summary.
Frequently asked questions
How does the Standard Deviation Calculator calculator work?
It uses standard algebraic, geometric, or statistical theorems to return exact values.
How accurate are the results?
All calculations use double-precision floating-point arithmetic for exact accuracy.
Can I copy the step-by-step work?
Yes, step-by-step solutions are formatted clearly for homework or documentation.
How are undefined operations handled?
The calculator flags mathematical errors such as division by zero or negative square roots.
Is this tool suitable for students and engineers?
Yes, it is designed for students, teachers, engineers, and scientists.
Does it support negative numbers and decimals?
Yes, all real numbers (positive, negative, fractions, and decimals) are supported.
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