Standard Deviation Calculator
Enter numbers separated by commas, spaces, or new lines. Choose population or sample.
Standard Deviation — Formula & Guide
Population SD (σ)
σ = √[ Σ(xᵢ − μ)² / N ]
Use when your data is the entire population.
Sample SD (s)
s = √[ Σ(xᵢ − x̄)² / (n−1) ]
Use when your data is a sample from a larger population (Bessel's correction).
Frequently Asked Questions
How do I calculate standard deviation by hand?
For data {2, 4, 4, 4, 5, 5, 7, 9}: (1) Find mean = (2+4+4+4+5+5+7+9)/8 = 5. (2) Subtract mean, square each: (2−5)²=9, (4−5)²=1×3=3, (5−5)²=0×2=0, (7−5)²=4, (9−5)²=16. (3) Sum = 9+3+3+3+0+0+4+16 = 38. Wait: (4−5)² = 1, three of them = 3. Sum = 9+1+1+1+0+0+4+16 = 32. (4) Divide by N−1=7 → 32/7 ≈ 4.57. (5) √4.57 ≈ 2.14.
Is a higher or lower standard deviation better?
Neither is inherently better — it depends on context. Low SD means consistent, predictable values (good for quality control or test scoring). High SD means high variability (sometimes desirable in investment returns meaning upside potential, but often indicates inconsistency).
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
- Mean = (2+4+4+4+5+5+7+9) / 8 = 5
- Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
- Population variance = 32 / 8 = 4
- SD = √4 = 2
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- Mean, Median & Mode Calculator
- Z-Score Calculator
- Confidence Interval Calculator
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References
- Pearson, K. (1894). "Contributions to the Mathematical Theory of Evolution." Phil. Trans. Royal Society A, 185.
- Freedman, D., Pisani, R. & Purves, R. (2007). Statistics (4th ed.). Norton.
