Variance Calculator
Calculate sample and population variance with step-by-step explanations
Enter Your Data
Dataset Type
Use sample variance when your data represents a subset of a larger population. Use population variance when you have complete data for the entire population.
Steps to Show
Understanding Variance
Low Variance
Data points are close to the mean
High Variance
Data points are spread out from the mean
Standard Deviation
Square root of variance, same units as data
Sample vs Population
Sample Variance (s²)
Uses N-1 (Bessel's correction) for unbiased estimation
Population Variance (σ²)
Uses N when you have complete population data
Quick Tips
Variance is always non-negative
Units are squared (e.g., cm²)
Standard deviation has same units as data
Higher variance = more data spread
Worked Example
Student Quiz Scores
Eight students' quiz scores:
Step 1: Calculate Mean
Mean = (5+5+5+7+8+8+9+9) ÷ 8 = 56 ÷ 8 = 7
Step 2: Calculate Squared Deviations
(5-7)² = 4, (5-7)² = 4, (5-7)² = 4, (7-7)² = 0
(8-7)² = 1, (8-7)² = 1, (9-7)² = 4, (9-7)² = 4
Sum of squared deviations = 22
Final Calculation
Population Variance
σ² = 22 ÷ 8 = 2.75
σ = √2.75 = 1.658
Sample Variance
s² = 22 ÷ (8-1) = 22 ÷ 7 = 3.143
s = √3.143 = 1.773
Understanding Variance
What is Variance?
Variance is a measure of how spread out data points are from their mean. It quantifies the variability or dispersion in a dataset. A higher variance indicates more spread, while a lower variance indicates data points are closer to the mean.
Why is Variance Important?
- •Measures data variability and spread
- •Foundation for standard deviation calculation
- •Used in statistical hypothesis testing
- •Essential for confidence intervals
Variance Formulas
Population Variance
Used when you have data for the entire population
Sample Variance
Used when estimating population variance from a sample
Bessel's Correction: Sample variance uses N-1 instead of N to provide an unbiased estimate of the population variance.
Applications of Variance
Quality Control
Monitor process consistency and detect anomalies
Finance
Measure investment risk and portfolio volatility
Research
Compare variability between different groups