Variance Calculator
Calculate population and sample variance with detailed statistics
σ² = Σ(xᵢ - μ)² / N
Population variance - use when you have data for the entire population
s² = Σ(xᵢ - x̄)² / (n-1)
Sample variance - use when you have a sample from a larger population
Minimum 2 numbers required
Quick Examples
Variance
=
Mean (μ/x̄)
Std Deviation
Count (n)
Sum (Σx)
Min
Max
Range
Sum of Squares
Step-by-Step Breakdown
Step 1: Your Data
Step 2: Calculate Mean
μ = () / =
Step 3: Calculate Deviations
| x | x - μ | (x - μ)² |
|---|---|---|
Step 4: Calculate Variance
= Σ(x - μ)² / =
Step 5: Standard Deviation (√Variance)
= √ =
Note: Step-by-step breakdown is hidden for datasets larger than 10 values for readability.
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About Variance Calculator
What is Variance?
Variance is a statistical measure that quantifies how far data points are spread from the mean. It measures the average of the squared differences between each data point and the mean. A higher variance indicates more spread in the data.
Population vs Sample Variance
Population Variance (σ²)
Used when you have data for the entire population:
σ² = Σ(xᵢ - μ)² / N
Where:
- σ² = population variance
- μ = population mean
- N = total number of values
- xᵢ = each value in the dataset
Sample Variance (s²)
Used when you have a sample from a larger population:
s² = Σ(xᵢ - x̄)² / (n-1)
Where:
- s² = sample variance
- x̄ = sample mean
- n = sample size
- Uses (n-1) for Bessel's correction
Relationship to Standard Deviation
Variance is the square of standard deviation:
- Variance = (Standard Deviation)²
- Standard Deviation = √Variance
While variance is useful in statistical calculations, standard deviation is often preferred for interpretation because it's in the same units as the original data.
How to Calculate Variance
- Find the mean (average) of your data
- Subtract the mean from each value
- Square each difference
- Find the average of squared differences (divide by N for population, n-1 for sample)
Example Calculation
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
- Mean = (2+4+4+4+5+5+7+9) / 8 = 5
- Differences from mean: -3, -1, -1, -1, 0, 0, 2, 4
- Squared differences: 9, 1, 1, 1, 0, 0, 4, 16
- Sum of squares = 32
- Population Variance = 32/8 = 4
- Sample Variance = 32/7 ≈ 4.57
When to Use Each Type
| Scenario | Use |
|---|---|
| Survey of all employees | Population Variance |
| Sample from a larger group | Sample Variance |
| Scientific measurements | Depends on context |
| Quality control | Usually Sample Variance |
Applications of Variance
- Finance - Measuring portfolio risk and volatility
- Quality Control - Process variation analysis
- Research - Statistical hypothesis testing
- Machine Learning - Feature scaling and normalization
- Weather - Climate variability studies
Tip: For most practical applications with sample data, use Sample Variance (s²) which provides an unbiased estimate of the population parameter.