Variance and Standard Deviation Calculator

Calculate population and sample variance and standard deviation from a dataset. Enter comma-separated values to find σ² (variance) and σ (standard deviation) with worked steps.

📊 Statistics📐 s² = Σ(xi - x̄)² / (n-1)🔢 Math
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Formula & Reference

VariableSymbolFormulaUnits
Variance and Standard Deviation Calculator—s² = Σ(xi - x̄)² / (n-1)squared units

Step-by-Step Examples

Example 1
Small Dataset

Data: 4, 7, 2, 9, 5. Find sample variance and SD.

  • Mean = (4+7+2+9+5)/5 = 27/5 = 5.4
  • SS = (4-5.4)² + (7-5.4)² + (2-5.4)² + (9-5.4)² + (5-5.4)²
  • = 1.96 + 2.56 + 11.56 + 12.96 + 0.16 = 29.2
  • s² = 29.2/(5-1) = 7.3
  • s = sqrt(7.3) = 2.702
✓ s² = 7.3, s = 2.702
Example 2
Temperature Data

Daily temperatures (°F): 72,75,68,80,71,79,66.

  • Mean = 73
  • SS = Σ(xi-73)² = 1+4+25+49+4+36+49 = 168
  • s² = 168/6 = 28.0
  • s = 5.29°F
  • Low variability — fairly consistent temperatures
✓ s = 5.29°F
Example 3
Population vs Sample

Same data: 4,7,2,9,5. Differences in calculation.

  • Sample: s² = SS/(n-1) = 29.2/4 = 7.30
  • Population: σ² = SS/N = 29.2/5 = 5.84
  • Sample variance is larger (Bessel's correction)
  • Use sample when data is a subset of a larger population
✓ Sample s²=7.3, Population σ²=5.84

Real-World Applications

📊
Risk Analysis
Variance measures investment risk. Higher variance = more unpredictable returns.
🏭
Quality Control
Six Sigma uses standard deviation to measure process variability and defect rates.
📊
Statistics
Variance is fundamental to ANOVA, regression, and many hypothesis tests.
🧪
Experimental Science
Variance from control and experimental groups compared with F-test in ANOVA.

Common Mistakes to Avoid

⚠️
Dividing by n instead of n-1 for sample data

For samples: use n-1 (Bessel's correction) to get an unbiased estimate of population variance. Only divide by N when you have the entire population.

⚠️
Confusing variance and SD

Variance = SD². Variance is in squared units (e.g., cm²). SD is in original units (e.g., cm). SD is more interpretable; variance is more useful mathematically.

⚠️
SD ≠ margin of error

SD describes spread of data values. SE = SD/√n describes precision of the mean estimate. These are very different and often confused.

Frequently Asked Questions

What is variance? ▾
The average squared deviation from the mean. Measures how spread out data is. Low variance: data clusters near mean. High variance: data is widely spread.
Why square the deviations? ▾
Squaring: 1) prevents positive and negative deviations from canceling, 2) penalizes larger deviations more heavily, 3) gives mathematical properties needed for further statistics.
Why n-1 for sample variance? ▾
Bessel's correction: using the sample mean (which is itself estimated from the data) underestimates the true spread. Dividing by n-1 corrects this bias and gives an unbiased estimator of population variance.
What is the coefficient of variation (CV)? ▾
CV = SD/mean × 100%. Relative measure of variability — useful for comparing datasets with different units or scales. CV=10% means SD is 10% of the mean.
What is the variance addition rule? ▾
For independent random variables: Var(X + Y) = Var(X) + Var(Y). For sums of independent measurements, variance adds up. This is why SD of n observations grows as sqrt(n).

Related Math Calculators

📊 Standard Deviation📊 Standard Error📊 Statistics📊 IQR📊 Correlation

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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