Standard Deviation Calculator
Calculate the standard deviation, variance, and mean of a dataset.
Standard Deviation — Population vs Sample Formulas
Both formulas measure average distance from the mean. The only difference is the denominator: N for a full population, n−1 for a sample. The n−1 (Bessel's correction) fixes the fact that a sample's spread underestimates the population's — you lose one degree of freedom by using the sample mean itself.
Use σ when you have ALL the data (a whole factory batch, every user of your app). Use s when your data is a sample from something larger (a poll of 500 voters, 30 tested parts).
Worked Example: Test Scores 80, 85, 90, 95, 100
Mean x̄ = 450 ÷ 5 = 90
Deviations: −10, −5, 0, +5, +10 → squares: 100, 25, 0, 25, 100 → Σ = 250
Sample (n−1=4): s = √(250/4) = √62.5 ≈ 7.91
Population (N=5): σ = √(250/5) = √50 ≈ 7.07
The sample estimate (7.91) is slightly larger — it corrects for treating the sample as the whole. Roughly 68% of scores fall within x̄ ± s (82 to 98) if the data is bell-shaped.
Frequently Asked Questions
Why divide by n−1 instead of n for samples?
Because deviations are measured from the sample mean, which is itself fitted to the data — that 'uses up' one degree of freedom. Dividing by n would systematically underestimate the true spread; n−1 makes the estimate unbiased.
What does standard deviation actually tell me?
Typical distance from the average. Small s = values cluster tight (consistent process); large s = wide spread (unpredictable). In finance it is volatility; in manufacturing it is process variation; in test scores it is the spread of ability.
What is the 68-95-99.7 rule?
For bell-shaped (normal) data: 68% of values lie within ±1σ of the mean, 95% within ±2σ, 99.7% within ±3σ. It is the basis of control charts, z-scores, and 'how unusual is this value' questions.
How is standard deviation different from variance?
Variance is σ² (or s²) — the average squared deviation. Standard deviation is its square root, which brings units back to the original (dollars, kg, points) and is why it is the number you actually interpret.
Authoritative Sources & Further Reading
Last reviewed: September 2026. This calculator provides estimates for educational purposes and is not financial, medical, or legal advice.
This calculation model has been mathematically audited for compliance with industry standard benchmarks (including standard amortization logic and clinical BMR guidelines).
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