Population Variance Calculator
Calculate population variance and standard deviation for complete datasets
Calculate Population Variance
Enter all population values separated by commas, spaces, or line breaks
Example Calculation
Population Dataset Example
Dataset: 2, 4, 6, 8, 10
Population Size (N): 5
Population Mean (μ): (2+4+6+8+10)/5 = 6
Step-by-Step Calculation
1. Calculate deviations: (2-6)=-4, (4-6)=-2, (6-6)=0, (8-6)=2, (10-6)=4
2. Square deviations: 16, 4, 0, 4, 16
3. Sum of squared deviations: 16+4+0+4+16 = 40
4. Population variance: σ² = 40/5 = 8.0
5. Standard deviation: σ = √8 = 2.828
Key Concepts
Population Variance
Average of squared deviations from mean
Population Mean
Average value of all data points
Population Size
Total number of data points
When to Use
When you have the complete population dataset
Quality control in manufacturing processes
Analyzing complete experimental results
Business analytics with complete datasets
Understanding Population Variance
What is Population Variance?
Population variance (σ²) measures how spread out data points are in a complete population. It's the average of the squared differences between each data point and the population mean. A higher variance indicates more spread in the data, while lower variance means data points cluster closer to the mean.
Key Characteristics
- •Always non-negative (≥ 0)
- •Measured in squared units of original data
- •Uses the entire population (N in denominator)
- •Smaller than sample variance for the same data
Formula Breakdown
σ² = Σ(xᵢ - μ)² / N
- σ²: Population variance (sigma squared)
- xᵢ: Individual data point
- μ: Population mean (mu)
- N: Total population size
- Σ: Sum of all terms
Note: Population variance divides by N, while sample variance divides by N-1 (Bessel's correction) to account for sampling bias.
Population vs Sample Variance
Population Variance (σ²)
- • Uses complete population data
- • Divides by N (population size)
- • Provides exact measure of variability
- • Smaller value than sample variance
Sample Variance (s²)
- • Uses subset of population data
- • Divides by N-1 (Bessel's correction)
- • Estimates population variance
- • Larger value to correct for bias
Applications
Quality Control
Measure consistency in manufacturing processes where all items can be tested.
Risk Assessment
Evaluate variability in financial portfolios or investment returns.
Research Analysis
Analyze complete experimental datasets or survey populations.