math ToolUpdated 2026-09-19
Standard Deviation Calculator
Measure the statistical dispersion and spread of data around the mean with sample and population standard deviation.
Formula & Mathematical Methodology
s = √(Σ(x − x̄)² ÷ (n − 1))
Find the mean, sum the squared deviations from the mean, divide by n-1 (for sample) or N (for population), and take the square root.
How to Use This Standard Deviation Calculator
- Enter numbers separated by commas.
- Select sample or population.
- Click Calculate for standard deviation and variance.
Step-by-Step Worked Example
Scenario: 10, 12, 23, 23, 16, 23, 21, 16
1. Calculate mean = 18
2. Sum squared diffs = 176
3. Divide by n - 1 (7) = 25.14 variance
4. Take square root = 5.01
Final Calculated Result: 5.0143
What Does Your Result Mean?
A low standard deviation indicates values cluster tightly near the mean; high deviation indicates broad variability.
Common Practical Applications:
- Financial risk assessment (volatility)
- Quality control in manufacturing
- Scientific experiment repeatability
Important Limitations & Assumptions:
- Sample standard deviation requires at least 2 observations.
Frequently Asked Questions
What is Bessel’s correction?
Using n-1 in the denominator instead of n compensates for bias when estimating population variance from a sample.
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