Calculate Range, Variance, SD with Step-by-Step Derivation
Enter numbers separated by commas
Results
Range
Variance
Std Dev
Mean
Step-by-Step Derivation
Formulas
Range = Max - Min
Population Variance σ² = Σ(x - μ)² / N
Sample Variance s² = Σ(x - x̄)² / (n-1)
Standard Deviation = √Variance
Range measures spread. Variance and standard deviation measure dispersion from the mean.
⚠Enter numbers separated by commas. Minimum 2 values required. Maximum 50 values.
What are Range, Variance & Standard Deviation?
These are measures of dispersion that describe how spread out data is. Range is simple, variance and standard deviation are more precise measures.
Range
Difference between maximum and minimum values. Simple but doesn't show distribution shape.
Variance
Average of squared differences from the Mean. Measures spread but in squared units.
Standard Deviation
Square root of variance. Most common measure of spread in original units.
Population vs Sample
Population uses all data (N). Sample uses n-1 to correct estimation bias.
💡 Example: Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5, Range = 7, Variance = 4, Std Dev = 2.
Applications
Quality ControlFinanceScienceEngineeringSocial Research
Frequently Asked Questions
What is the difference between range, variance, and standard deviation?▼
Range is the difference between max and min (simple but limited). Variance measures average squared deviation from mean. Standard deviation is square root of variance (same units as data).
What is population vs sample standard deviation?▼
Population uses N in denominator (entire dataset). Sample uses n-1 (Bessel's correction) to correct bias when estimating population parameters from a sample.
Why use n-1 for sample variance?▼
Using n-1 (degrees of freedom) corrects for bias when estimating population variance from a sample. Without it, the estimate would systematically underestimate the true variance.
What does standard deviation tell us?▼
Standard deviation measures the spread or dispersion of data. A small standard deviation means data points are close to the mean; a large standard deviation means data is spread out.
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