Standard Deviation Calculator
Calculate the mean, variance, and standard deviation for a set of numbers. Enter values separated by commas.
Enter Your Data
Statistical Results
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data points
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sum / count
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divides by N-1
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divides by N
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s squared
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σ squared
💡 Which one to use? Use Sample standard deviation if your data is a subset of a larger group. Use Population if your data represents the entire group you are studying.
What is Standard Deviation?
Standard deviation is a measure of how spread out numbers are from the mean (average). A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
The Formulas
Population Standard Deviation (σ):
σ = √[ Σ(xi - μ)² / N ]
Sample Standard Deviation (s):
s = [ Σ(xi - x̄)² / (N - 1) ]
σ = √[ Σ(xi - μ)² / N ]
Sample Standard Deviation (s):
s = [ Σ(xi - x̄)² / (N - 1) ]
Step-by-Step Calculation
- Find the Mean: Add all numbers together and divide by the count.
- Calculate Deviations: Subtract the mean from each number.
- Square the Deviations: Multiply each deviation by itself (to remove negatives).
- Find the Variance: Average the squared deviations (divide by N for population, N-1 for sample).
- Take the Square Root: The square root of the variance is the standard deviation.
Sample vs. Population
| Feature | Sample (s) | Population (σ) |
|---|---|---|
| Denominator | N - 1 (Bessel's correction) | N |
| Use Case | Surveying a subset | Census of entire group |
| Result Size | Slightly larger (more conservative) | Slightly smaller |
The Empirical Rule (68-95-99.7)
For a normal distribution (bell curve):
- 68% of data falls within 1 standard deviation of the mean.
- 95% of data falls within 2 standard deviations.
- 99.7% of data falls within 3 standard deviations.
References
- Khan Academy. "Standard Deviation." 2024.
- Investopedia. "Standard Deviation in Finance." 2024.