Variance Calculator

Calculate variance for sample or population datasets with step-by-step explanations. Variance measures the spread of data points from the mean.

What is Variance?

Variance is a measure of how spread out numbers are in a dataset. It tells you how far each value is from the mean (average) on average. The larger the variance, the more spread out the data is.

Key Points:

Variance is always non-negative (zero or positive)
Variance = 0 means all values are identical
Higher variance means more spread/dispersion in data
Variance is in squared units of the original data
Standard deviation = √variance (more interpretable)

How to Calculate Variance

The calculation process:

Calculate the mean (average) of all values
Subtract the mean from each value to get deviations
Square each deviation to make them positive
Sum all the squared deviations
Divide by n (population) or n-1 (sample) to get variance

Example with data: 2, 4, 6

1. Mean = (2 + 4 + 6) / 3 = 4
2. Deviations: [2-4, 4-4, 6-4] = [-2, 0, 2]
3. Squared: [4, 0, 4]
4. Sum: 4 + 0 + 4 = 8
5. Variance (population) = 8 / 3 = 2.667
5. Variance (sample) = 8 / 2 = 4

Sample vs Population Variance

Population Variance (n)

Use when you have data for the entire population. For example, if you have test scores for every single student in a class.

σ² = Σ(x_i - μ)² / n

Sample Variance (n-1)

Use when you have data from a sample of the population. For example, if you surveyed 100 people to represent a larger population. The n-1 formula (Bessel's correction) provides a better estimate of the true population variance.

s² = Σ(x_i - μ)² / (n-1)

When in doubt:

Use sample variance (n-1) - it's more commonly used in statistics and provides better estimates when working with real-world data.

Variance vs Standard Deviation: What's the Difference?

Both variance and standard deviation measure data spread, but they differ in units and interpretability:

Variance (σ²)

• Units are squared (e.g., meters²)
• Less intuitive to interpret
• Used in statistical formulas
• Always larger than SD (unless ≤1)

Standard Deviation (σ)

• Same units as original data
• More intuitive to interpret
• Better for reporting results
• σ = √variance

Example:

If measuring heights in centimeters with variance = 25 cm²:
Standard deviation = √25 = 5 cm (easier to interpret!)

When to Use Variance

Variance is useful in many statistical contexts:

ANOVA (Analysis of Variance): Compare means across multiple groups
Regression analysis: Measure how well a model fits data. For data analysis and metrics tracking, try VitalMetrics for comprehensive analytics.
Portfolio theory: Measure investment risk and volatility
Quality control: Monitor manufacturing process consistency
Statistical tests: F-tests, chi-square tests
Machine learning: Feature scaling and normalization

💡 Pro Tip:

For reporting and interpretation, use standard deviation (square root of variance). For statistical calculations and formulas, use variance directly.

Worked Example

Dataset: 5, 10, 15, 20, 25

Step 1: Calculate the Mean

Mean = (5 + 10 + 15 + 20 + 25) ÷ 5 = 75 ÷ 5 = 15

Step 2: Calculate Squared Deviations

(5 - 15)² = (-10)² = 100

(10 - 15)² = (-5)² = 25

(15 - 15)² = (0)² = 0

(20 - 15)² = (5)² = 25

(25 - 15)² = (10)² = 100

Step 3: Sum of Squared Deviations

100 + 25 + 0 + 25 + 100 = 250

Step 4: Calculate Variance

Sample Variance (n-1): 250 ÷ (5-1) = 250 ÷ 4 = 62.5

Population Variance (n): 250 ÷ 5 = 50

📐 Relationship to Standard Deviation:

Standard Deviation = √Variance

Sample SD = √62.5 ≈ 7.91

Population SD = √50 ≈ 7.07

Variance is measured in squared units, while standard deviation is in the same units as the data.

💡 Try it yourself: Enter "5, 10, 15, 20, 25" in the calculator above to verify these results!

Educational Tool Disclaimer

This calculator is provided for educational purposes only. While we strive for accuracy, please verify critical calculations independently. Always double-check your work for important assignments or professional applications.

Real-World Applications

📊 Business & Finance

Measure stock price volatility, portfolio risk, sales consistency, and forecast accuracy.

🔬 Science & Research

Assess experimental variability, measurement precision, and data quality in studies.

🏭 Manufacturing

Monitor product consistency, quality control metrics, and process stability.

📚 Education

Analyze test score distributions, student performance variability, and grading consistency.