Math Calculator
Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, sum, and more from any dataset. Full step-by-step solution with deviation table — no spreadsheet needed.
Standard Deviation Calculator
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Dataset Type
Dataset (comma or space separated)
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Formulas
Mean
x̄ = Σx ÷ n
Sample SD
s = √(Σ(x−x̄)² ÷ (n−1))
Population SD
σ = √(Σ(x−x̄)² ÷ n)
Variance
s² = Σ(x−x̄)² ÷ (n−1)
Interpretation Guide
| SD Value | Meaning |
|---|---|
| Low SD | Data clustered near mean |
| High SD | Data spread widely |
| SD = 0 | All values identical |
| ±1 SD | ~68% of normal data |
| ±2 SD | ~95% of normal data |
| ±3 SD | ~99.7% of normal data |
Frequently Asked Questions
Step 1: Find the mean. Step 2: Subtract mean from each value and square the result. Step 3: Sum all squared deviations. Step 4: Divide by n−1 (sample) or n (population) for variance. Step 5: Take the square root. Use the Standard Deviation Calculator above for instant results.
Sample SD (s) divides by n−1 (Bessel's correction) to give an unbiased estimate of population spread when you only have a sample. Population SD (σ) divides by n and is used when you have data for every member of the group. Sample SD is larger than population SD for the same dataset.
Standard deviation measures how spread out values are around the mean. A small SD means values cluster tightly around the mean. A large SD means values are dispersed widely. In a normal distribution, 68% of values fall within ±1 SD and 95% within ±2 SD of the mean.
Variance is the square of standard deviation (SD² = variance). Variance is computed first (average squared deviation from mean), then square-rooted to give SD. Variance uses squared units (making it harder to interpret), while SD is in the same units as the original data.
There's no universally "good" SD — it depends on context. A coefficient of variation (CV = SD/mean) below 15% is often considered low variability. For test scores, an SD of 10–15 is typical. For stock returns, a lower SD means less volatility. Interpret SD relative to the mean and domain.
For normally distributed data: 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. Values beyond 3 SD are extremely rare (0.3%). This rule applies specifically to bell-shaped (normal) distributions, not all datasets.
MAD = average of |x − mean|. SD = square root of average squared deviations. SD gives more weight to outliers (because of squaring) and is used in most statistical tests. MAD is more robust to outliers. SD is generally preferred in inferential statistics and hypothesis testing.
CV = (Standard Deviation ÷ Mean) × 100%. It expresses SD as a percentage of the mean, making it useful for comparing variability between datasets with different units or scales. A CV under 15% indicates low variability; above 30% indicates high variability relative to the mean.