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

1. Find the mean (average) of your data
2. Subtract the mean from each value
3. Square each difference
4. 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

1. Mean = (2+4+4+4+5+5+7+9) / 8 = 5
2. Differences from mean: -3, -1, -1, -1, 0, 0, 2, 4
3. Squared differences: 9, 1, 1, 1, 0, 0, 4, 16
4. Sum of squares = 32
5. Population Variance = 32/8 = 4
6. Sample Variance = 32/7 ≈ 4.57

When to Use Each Type

Applications of Variance

1. Finance - Measuring portfolio risk and volatility
2. Quality Control - Process variation analysis
3. Research - Statistical hypothesis testing
4. Machine Learning - Feature scaling and normalization
5. 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.