Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, and other statistical measures with detailed explanations
Data Input
Enter numeric values only
Results
Enter data values and click "Calculate Statistics" to see the results
Understanding Standard Deviation
What is Standard Deviation?
Standard deviation is a measure of how spread out data points are from the mean (average). A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.
Sample: σ = √[Σ(xi - x̄)² / (n-1)]
Population: σ = √[Σ(xi - μ)² / n]
Sample vs Population
The choice between sample and population standard deviation depends on your data:
- Sample (n-1): When your data represents a sample from a larger population
- Population (n): When your data represents the entire population of interest
- Bessel's correction: Sample uses n-1 to provide unbiased estimates
Step-by-Step Calculation Process
- 1. Calculate the mean: Add all values and divide by the count
- 2. Find deviations: Subtract the mean from each value
- 3. Square deviations: Square each deviation to remove negative signs
- 4. Sum squared deviations: Add all squared deviations together
- 5. Divide by n or n-1: Use n for population, n-1 for sample
- 6. Take square root: The result is the standard deviation
Applications
- • Quality control
- • Risk assessment
- • Scientific research
- • Financial analysis
- • Performance evaluation
Interpretation
- • 68% within 1σ of mean
- • 95% within 2σ of mean
- • 99.7% within 3σ of mean
- • Higher σ = more variability
- • Lower σ = more consistency
Related Measures
- • Variance (σ²)
- • Coefficient of variation
- • Z-scores
- • Range and IQR
- • Mean absolute deviation
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