Standard Deviation Calculator
Enter numbers to instantly calculate mean, variance and standard deviation (σ and s). Step-by-step display available.
Standard Deviation: Measure the Spread in Your Data
Standard deviation is a statistical measure that shows how far values in a dataset deviate from the mean. Small standard deviation = values are close together; large standard deviation = values are spread over a wide range. This tool calculates population (σ) and sample (s) standard deviation, variance, and mean step by step from a comma- or space-separated list of numbers.
Standard Deviation Formula
σ (population) = √[ Σ(xᵢ − x̄)² / n ]
s (sample) = √[ Σ(xᵢ − x̄)² / (n − 1) ]
x̄ = mean, xᵢ = each value, n = count of data points. The (n−1) divisor in the sample formula is known as "Bessel's correction" and ensures the sample variance is an unbiased estimate.
How to Use
Entering Numbers
Enter numbers separated by commas, spaces, semicolons, or line breaks. You may use either a period or comma as the decimal separator. Example: 2.5, 3.8, 4.1, 2.9, 5.0
Interpreting Results
If you have the complete dataset (e.g. all employee salaries at a company), use population σ. If you are generalising from a sample (e.g. a survey), use sample s.
Step-by-Step Display
After calculating, expand the "Step-by-Step Calculation" section to see the (xᵢ − x̄)² differences for each value. This feature is designed for students and educational use.
Example Calculation
Dataset: 2, 4, 4, 4, 5, 5, 7, 9 (n=8)
- Mean x̄ = (2+4+4+4+5+5+7+9) / 8 = 5
- Sum of squares = (2−5)²+(4−5)²+(4−5)²+(4−5)²+(5−5)²+(5−5)²+(7−5)²+(9−5)² = 9+1+1+1+0+0+4+16 = 32
- Population variance σ² = 32/8 = 4 → σ = 2
- Sample variance s² = 32/7 ≈ 4.571 → s ≈ 2.138
Where Is Standard Deviation Used?
- Finance: Return volatility of stocks or investment funds; the larger σ is, the higher the risk.
- Quality control: Checking whether product dimensions in manufacturing fall within tolerance limits.
- Education: Distribution of class grades; low σ means scores are clustered, high σ means they are spread widely.
- Scientific research: Measurement uncertainty and experimental error analysis.