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σ²

Variance: σ² or s²

Variance measures spread: mean of squared deviations. Population σ² divides by N; sample s² uses Bessel correction (n−1). σ = √σ².

Concept Fundamentals
Population variance
σ²
Sample variance
s²
Population mean
μ
Sample mean
x̄

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σ² = (1/N)Σ(x_i−μ)². s² = (1/(n−1))Σ(x_i−x̄)². Bessel correction: n−1 for unbiased sample estimate. Standard deviation σ = √σ².

Key quantities
Population variance
σ²
Key relation
Sample variance
s²
Key relation
Population mean
μ
Key relation
Sample mean
x̄
Key relation

Ready to run the numbers?

Why: Variance quantifies spread around the mean. ANOVA uses between-group variance. Risk in finance. Quality control.

How: Compute mean: μ or x̄. For each value: (x_i − μ)². Sum and divide by N (population) or n−1 (sample).

σ² = (1/N)Σ(x_i−μ)². s² = (1/(n−1))Σ(x_i−x̄)².Bessel correction: n−1 for unbiased sample estimate.
Sources:Khan Academy StatisticsNIST

Run the calculator when you are ready.

Calculate VarianceEnter data values
variance.sh
CALCULATED
$ variance --values "2, 4, 6, 8, 10"
Mean (μ)
6
s²
10
σ = √σ²
3.162278
n
5
Variance Calculator
s² = 10
Mean: 6 | σ: 3.162278 | n = 5
numbervibe.com
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Summary Metrics

Squared Deviations Contribution

📐 Step-by-Step Breakdown

INPUTS
n
5
MEAN
Mean (μ)
6
\text{Sigma} x_i / n
DEVIATIONS
Squared deviations
(2 − 6)², (4 − 6)², (6 − 6)²...
Sum of squared deviations
40
\text{Sigma} (x_i - \text{mu} )^{2}
Divisor
4
n − 1
RESULT
s²
10
\text{sigma} ^{2} = \text{Sigma} (x_i - \text{mu} )^{2} / divisor
σ = √(variance)
3.162278
ext{Standard} ext{deviation}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

σ²

Variance = mean of squared deviations

— Definition

📐

s² uses n−1 for unbiased estimate

— Bessel

📋 Key Takeaways

  • • Variance = average of squared deviations from the mean: σ² = (1/N)Σ(x_i − μ)²
  • • Population σ²: divide by N. Sample s²: divide by (n−1) — Bessel correction
  • • σ = √(variance) — standard deviation has same units as data; variance has squared units
  • • Variance = 0 iff all values are identical
  • • Var(aX + b) = a²Var(X) — adding constant doesn't change variance

💡 Did You Know?

📐Variance has squared units (e.g. m² if data in meters).Source: Units
📊σ² = E[(X−μ)²] for probability distributions.Source: Expectation
⚖️Bessel correction (n−1) gives unbiased sample variance.Source: Estimation
📈In finance, variance measures risk and volatility.Source: Quantitative Finance
🔬ANOVA decomposes variance into between-group and within-group.Source: Statistics
💻Computational formula: Var(X) = E[X²] − (E[X])².Source: Efficient Calculation

📖 How Variance Works

Compute the mean μ, then for each value find (x_i − μ)², sum them, and divide by N (population) or n−1 (sample). Squaring ensures all deviations are positive and penalizes large deviations more. Outliers inflate variance significantly.

σ² = (1/N) Σ(x_i − μ)² → σ = √σ²

📝 Worked Example: 2, 4, 6, 8, 10

Step 1: Mean μ = (2+4+6+8+10)/5 = 6

Step 2: Squared deviations: (2−6)²=16, (4−6)²=4, (6−6)²=0, (8−6)²=4, (10−6)²=16

Step 3: Sum = 40. Sample: 40/(5−1) = 10

Result: s² = 10, s = √10 ≈ 3.16

🚀 Real-World Applications

📊 Finance

Portfolio risk, volatility, option pricing.

🔬 Research

Measurement error, reproducibility.

🏭 Quality Control

Process variability, Six Sigma.

📈 Machine Learning

Feature scaling, regularization.

🎲 Probability

Distribution spread, risk models.

📉 Economics

Income inequality, Gini coefficient.

⚠️ Common Mistakes to Avoid

  • Using population when you have a sample: Use n−1 for unbiased estimate.
  • Reporting variance when std dev is expected: Variance has squared units; std dev is in original units.
  • Ignoring outliers: Squaring amplifies outliers; consider robust measures.
  • Comparing variance across different units: Use coefficient of variation instead.

🎯 Expert Tips

💡 Population vs Sample

Population: entire group. Sample: subset. Use n−1 for sample variance.

💡 Computational Formula

Var(X) = E[X²] − (E[X])². More efficient for large datasets.

💡 Outliers

Squaring amplifies outliers. MAD is more robust.

💡 Linear Transform

Var(aX+b) = a²Var(X). Scaling changes variance; shifting does not.

📊 Reference Table

TypeDivisorSymbol
PopulationNσ²
Samplen − 1s²

❓ FAQ

Variance vs standard deviation?

Variance = σ². Std dev = σ = √(variance). Std dev has same units as data.

Population vs sample?

Population: entire group (divide by N). Sample: subset (divide by n−1 for unbiased estimate).

Why squared deviations?

Squaring makes all positive and penalizes large deviations more. Sum of unsquared deviations is always 0.

When variance = 0?

All values are identical. No spread.

Units of variance?

Squared units of data. E.g. meters → m². Use std dev for original units.

What is Bessel correction?

Using n−1 instead of n for sample variance gives an unbiased estimate of population variance.

📌 Summary

Variance measures the average squared deviation from the mean. It has squared units; take the square root for standard deviation. Use population formula when you have all data; sample formula (n−1) for subsets. Variance is fundamental in statistics, finance, and machine learning.

⚠️ Disclaimer: For very large datasets, consider computational formulas. Results are for educational purposes.

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