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Variance Calculator

Calculate population or sample variance with clear steps and formula. Paste data, choose mode/precision, get variance, mean, sum of squares and standard deviation.

Calculate Variance

Data (comma or space separated)
Mode
Precision (decimals)

Your Results

Count (n)
8
Mean (x̄)
5.00
Variance (σ²)
4.00
Std. Deviation (σ)
2.00

What is Variance?

Variance measures the average squared deviation from the mean. It quantifies how spread out the data is.

Formulas

Population (σ²): Σ(xᵢ − x̄)² / n
Sample (s²): Σ(xᵢ − x̄)² / (n − 1)

How to Calculate Variance

Steps

1) Compute the mean x̄
2) For each value, compute squared deviation (xᵢ − x̄)²
3) Sum all squared deviations
4) Divide by n (population) or (n − 1) (sample)

Example

Data: 2, 4, 4, 4, 5, 5, 7, 9

Mean x̄ = 5, Σ(xᵢ−x̄)² = 32

Population variance σ² = 32/8 = 4; Sample variance s² = 32/7 ≈ 4.5714

Important Considerations

⚠️ Data & Outliers

Outliers can heavily impact variance because deviations are squared.

🔢 Units & Scale

Variance depends on scale (units), unlike standard deviation normalization.

  • • Converting units changes variance magnitude
  • • Compare data in the same units
  • • Prefer std. deviation for interpretability
📊 Sample vs Population

Use sample mode when inferring to a broader population.

  • • Sample uses (n − 1) denominator
  • • Population uses n
  • • Small samples: sample variance more conservative
🧮 Numerical Stability

Large numbers may cause floating point issues; increase precision if needed.

  • • Prefer centered calculations Σ(xᵢ − x̄)²
  • • Avoid subtracting large close numbers
  • • Round display only; keep internal precision
📐 Interpretation

Variance is in squared units; take square root for standard deviation.

  • • Use SD for more intuitive spread
  • • Compare groups with similar scales
  • • Consider robust metrics if outliers exist

Tips

Clean your data first

Remove obvious typos and convert all values to the same unit before calculating variance.

Choose the right mode

Use population for full datasets; use sample when data is a subset.

Report precision clearly

Round for display only; keep full precision internally for calculations.

Use SD for communication

Standard deviation is easier to interpret; include both if helpful.

Example Cases

Case 1: Population variance example

Data: 2, 4, 4, 4, 5, 5, 7, 9
Mean: 5
σ²: 4

Classic textbook dataset demonstrating population variance.

Case 2: Sample variance example

Data: 2, 4, 4, 4, 5, 5, 7, 9
Mean: 5
s²: 32/7 ≈ 4.5714

Sample mode divides by (n − 1) to reduce bias.

Frequently Asked Questions

What is variance?
Variance measures the average squared deviation from the mean. It indicates how spread out the data is.
What's the difference between population and sample variance?
Population variance divides by n. Sample variance uses the unbiased estimator dividing by (n − 1). Use sample variance when your data is a sample of a larger population.
How many numbers do I need?
At least two values are required to compute a meaningful variance. A single value would yield zero spread but is not informative.
Why is variance in squared units?
Deviations are squared before averaging, so the variance unit is squared (e.g., cm²). Take the square root to get standard deviation with original units.
Do outliers affect variance?
Yes. Because deviations are squared, extreme values have a large effect. Consider robust measures or inspect data for outliers.

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