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How to Find Variance

Learn the formula, common mistakes, interactive calculator, and real-life applications.

Master measuring data spread with step-by-step guidance.

Quick Formula
The essential formulas you need to know
Variance = Average of Squared Deviations
Two different formulas:
Population
σ² = Σ(x - μ)² / n
Divide by n
Sample
s² = Σ(x - x̄)² / (n-1)
Divide by n-1
Population Example
Data: 2, 4, 6
Mean: 4
Deviations²: 4, 0, 4
Variance: 8 ÷ 3 = 2.67
Sample Example
Data: 2, 4, 6
Mean: 4
Deviations²: 4, 0, 4
Variance: 8 ÷ 2 = 4
Interactive Calculator
Enter numbers to calculate variance instantly

Use when you have data for the entire population

Press Enter or click Add to include the number

Practice Problems
Test your understanding with these variance calculation problems
Problem 1 of 4Score: 0/0

Find the population variance of: 2, 4, 6, 8

Common Mistakes to Avoid
Learn from these frequent errors to get accurate results
Confusing population and sample variance formulas
Population variance divides by n, sample variance divides by n-1 for unbiased estimation
Example: For data 1,2,3: Population σ² = 2/3 = 0.67, Sample s² = 2/2 = 1
Forgetting to square the deviations
Variance requires squared deviations from the mean, not just absolute deviations
Example: For deviation -2: Use (-2)² = 4, not |-2| = 2
Using the wrong mean in calculations
Always use the sample mean of your data, not a theoretical or assumed mean
Example: For data 3,5,7: Use mean = 5, not an assumed mean like 6
Confusing variance with standard deviation
Variance is σ², standard deviation is σ. Standard deviation = √variance
Example: If variance = 9, then standard deviation = √9 = 3, not 9
How the Formula Works
Understanding the mathematical principle behind variance

The Spread Measurement Concept

Variance measures how spread out data points are from the mean. It quantifies the average squared distance from each data point to the mean:

Population Variance Example: 2, 4, 6
Step 1: Find mean → (2 + 4 + 6) ÷ 3 = 4
Step 2: Find deviations → (2-4)², (4-4)², (6-4)²
Step 3: Calculate → 4, 0, 4
Variance = (4 + 0 + 4) ÷ 3 = 2.67
Sample Variance: Same data
Use n-1 in denominator for unbiased estimate
Sample Variance = 8 ÷ 2 = 4

Variance is always non-negative. Higher variance means data points are more spread out from the mean, while lower variance indicates data points cluster closer to the mean.

Real-World Applications
See how variance calculations are used in everyday life

Quality Control

Measure consistency in manufacturing processes and product specifications

Example: Widget weights: 100g, 102g, 98g, 101g → Low variance = consistent quality

Financial Risk

Assess investment risk by measuring return volatility and portfolio variance

Example: Stock returns: 5%, 8%, 2%, 10% → Higher variance = higher risk

Performance Analysis

Evaluate consistency in sports, testing, or any measurable performance

Example: Test scores: 85, 87, 83, 89 → Low variance = consistent performance

Scientific Research

Measure experimental precision and data reliability in research studies

Example: Lab measurements: 2.1, 2.3, 1.9, 2.2 → Variance shows measurement precision
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