Standard Deviation Calculator
Standard Deviation Calculator
Free online tool to calculate standard deviation, variance, and mean with step-by-step explanations.
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Standard Deviation Calculator: Measure Data Spread Instantly
Calculate standard deviation (σ), variance, mean, and count — with full step-by-step breakdown. Perfect for stats, finance, science, or quality control. Works for population or sample data.
Example: [85, 90, 88, 92, 87] → Mean = 88.4 | σ ≈ 2.88 (low spread).
Key Formulas Explained
Population σ
σ = √[Σ(x - μ)² / N]
μ = mean, N = total count
Sample s (Bessel)
s = √[Σ(x - x̄)² / (n - 1)]
Use for surveys, experiments
Variance
σ² = Σ(x - μ)² / N
Squared spread
Mean
μ = Σx / N
Center of data
Step-by-Step Calculation
| Data | x - μ | (x - μ)² |
|---|---|---|
| 85 | -3.4 | 11.56 |
| 90 | +1.6 | 2.56 |
| 88 | -0.4 | 0.16 |
| 92 | +3.6 | 12.96 |
| 87 | -1.4 | 1.96 |
| Σ = 442 | Σ = 0 | Σ = 29.2 |
Mean (μ) = 442 / 5 = 88.4
Variance = 29.2 / 5 = 5.84
σ = √5.84 ≈ 2.88
Real-World Applications
Finance
Stock volatility: High σ = high risk
Manufacturing
Quality control: σ < 1% = consistent parts
Science
Experiment error: Low σ = reliable results
Education
Test scores: σ shows class consistency
Pro Stats Tips
Best Practices
- Use sample s for surveys (n-1)
- Remove outliers before final calc
- Pair with mean for full insight
Common Mistakes
- Using N instead of n-1 in samples
- Ignoring units (σ in same unit as data)
- Confusing σ with variance
Why Trust This Calculator?
Accuracy: Uses NIST & ISO 3534-1 formulas — same as Excel, SPSS, and R.
Privacy: No data stored. Results vanish instantly.
Speed: JS-powered, works offline (PWA-ready).
Frequently Asked Questions
Sample: Divide by n-1 (unbiased estimate)