Variance Calculator

What is Variance?

Variance measures how far a set of numbers is spread out from their average (mean). It's calculated by averaging the squared differences from the mean.

• Low variance: Data points are close to the mean (consistent)
• High variance: Data points are spread out (variable)
• Zero variance: All values are identical

Population Variance (σ²)

Use when you have all possible data points:

σ² = Σ(x - μ)² / N

• μ (mu) = population mean
• N = total number of data points
• Example: Test scores of entire class

Sample Variance (s²)

Use when you have a sample from a larger population:

s² = Σ(x - x̄)² / (n-1)

• x̄ (x-bar) = sample mean
• n-1 = degrees of freedom (Bessel's correction)
• Dividing by n-1 instead of n gives an unbiased estimate
• Example: Survey of 100 people from a city of 1 million

Why Use n-1 for Samples?

Bessel's Correction: When using a sample to estimate population variance, dividing by n tends to underestimate the true variance. Dividing by (n-1) corrects this bias.

Variance vs. Standard Deviation

Variance (σ² or s²)	Standard Deviation (σ or s)
Squared units (e.g., cm²)	Original units (e.g., cm)
Harder to interpret	Easier to interpret
Used in formulas (ANOVA, etc.)	Used for practical interpretation
Always positive	Always positive
s² or σ²	s = √s² or σ = √σ²

Sum of Squares (SS)

The sum of squared deviations from the mean:

SS = Σ(x - μ)²

Variance is the average of the sum of squares:

• Population: σ² = SS / N
• Sample: s² = SS / (n-1)

Step-by-Step Example

Calculate variance for data: [2, 4, 4, 4, 5, 5, 7, 9]

1. Calculate mean: (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
2. Calculate deviations: [-3, -1, -1, -1, 0, 0, 2, 4]
3. Square each deviation: [9, 1, 1, 1, 0, 0, 4, 16]
4. Sum squared deviations: 9+1+1+1+0+0+4+16 = 32
5. Sample variance: 32 / (8-1) = 32 / 7 ≈ 4.57
6. Population variance: 32 / 8 = 4

Real-World Applications

• Finance: Portfolio variance measures investment risk
• Quality Control: Process variance shows manufacturing consistency
• Psychology: Test score variance shows group homogeneity
• Biology: Genetic variance within populations
• Weather: Temperature variance indicates climate stability
• Economics: Income variance measures inequality

When to Use Variance vs. Standard Deviation

• Use Variance: In statistical formulas (ANOVA, regression), when combining variances
• Use Standard Deviation: For interpretation, reporting, comparing to original data

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