Standard Deviation Calculator
A professional-grade analyzer for measuring data dispersion.
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Standard Deviation Calculator
Standard deviation measures how spread out data is from the mean. Whether you're analyzing test scores, stock prices, or experimental results, this calculator computes standard deviation, variance, and key statistics from your data set instantly.
How to Use This Calculator
- Enter your data values separated by commas, spaces, or new lines.
- Select population SD (σ) or sample SD (s).
- Click Calculate to get: mean, standard deviation, variance, min, max, range, and count.
Standard Deviation Formulas
Population SD (σ): σ = √[Σ(xᵢ − μ)² / N]
Sample SD (s): s = √[Σ(xᵢ − x̄)² / (N−1)]
Use population SD when you have data for every member of the group. Use sample SD when your data is a sample from a larger population (the extra −1 in the denominator is Bessel's correction).
Example Calculation
Data: 4, 7, 13, 16 | N = 4
- Mean (x̄): (4+7+13+16)/4 = 10
- Deviations: −6, −3, 3, 6
- Squared deviations: 36, 9, 9, 36
- Variance (population): (36+9+9+36)/4 = 22.5
- SD (population): √22.5 ≈ 4.74
- SD (sample): √(90/3) = √30 ≈ 5.48
Interpreting Standard Deviation
- Low SD: Data is clustered tightly around the mean (consistent).
- High SD: Data is spread widely from the mean (variable).
- Empirical rule (normal distribution): 68% within 1SD, 95% within 2SD, 99.7% within 3SD of the mean.
Common Mistakes to Avoid
- Using population SD for sample data — Using N instead of N−1 underestimates the true population variability. Use sample SD for survey data, experiment results, and any subset of a larger population.
- Confusing SD with variance — Variance is SD squared. SD has the same units as the original data; variance is in squared units. SD is usually more interpretable.
- Treating SD as the error — SD describes spread of the data. Standard Error (SE = SD/√N) describes uncertainty about the mean estimate.
Frequently Asked Questions
What is a good standard deviation?
It depends entirely on context and scale. A SD of 5 is large for a 0–10 rating scale but tiny for housing prices in the millions. Compare SD to the mean using the coefficient of variation (CV = SD/mean × 100%).
How is standard deviation related to normal distribution?
In a normal distribution, SD determines the bell curve's width. The 68-95-99.7 rule (empirical rule) tells you what percentage of data falls within 1, 2, or 3 standard deviations of the mean.
What is z-score and how does it use SD?
A z-score measures how many standard deviations a data point is from the mean: z = (x − μ)/σ. See our Z-Score Calculator.
Conclusion
Standard deviation is the most important single measure of data variability. Use this calculator for any data analysis task — academic, professional, or personal — and get the full suite of descriptive statistics in one click.
Related: Mean Median Mode Calculator | Z-Score Calculator | Statistics Calculator | Variance Calculator
Expert Note
If your dataset is very small, the Sample standard deviation will be significantly higher than the Population version due to "Bessel's Correction" (the N-1 factor), which accounts for the uncertainty in small samples.
