Variance Calculator
Variance & Standard Deviation Calculator
Calculate population variance, sample variance, standard deviation, mean, and other statistics. Step-by-step computation with formulas.
Population Variance
Use when you have all data
Sample Variance
Use when data is a sample
Variance Analysis Results
Variance Type:
Population Variance (σ²)
Formula Used:
σ² = Σ(x - μ)² / N
Interpretation:
Average squared deviation from mean
Data Spread:
Moderate variability
Deviation Analysis:
Data Visualization:
Points show spread around mean. Longer deviations = higher variance.
Step-by-Step Calculation:
Variance measures how far data points spread out from their mean. Higher variance = more spread.
What is Variance?
Variance is a statistical measure that quantifies the spread or dispersion of a set of data points around their mean (average) value. It represents the average of the squared differences from the mean. Variance is always non-negative, with higher values indicating greater variability in the data.
Types of Variance
Population Variance (σ²)
Use when you have data for entire population
Denominator: N (population size)
Sample Variance (s²)
Use when data is a sample from population
Denominator: n-1 (Bessel's correction)
Standard Deviation
Square root of variance
In original units of data
Mean Absolute Deviation
Alternative measure of spread
Uses absolute values instead of squares
Variance Formulas Explained
1. Population Variance Formula
Used when you have data for the entire population:
Where:
• σ² = Population variance
• N = Total number of data points
• xᵢ = Individual data point
• μ = Population mean = (Σxᵢ)/N
• (xᵢ - μ) = Deviation from mean
2. Sample Variance Formula
Used when data is a sample from a larger population:
Where:
• s² = Sample variance
• n = Sample size
• x̄ = Sample mean = (Σxᵢ)/n
• n-1 = Degrees of freedom (Bessel's correction)
3. Computational Formulas (Easier for Calculation)
Step-by-Step Variance Calculation
Example: Calculate Variance for [10, 12, 14, 16, 18]
- Step 1: Calculate Mean
μ = (10 + 12 + 14 + 16 + 18) / 5 = 70 / 5 = 14 - Step 2: Calculate Deviations
(10-14) = -4, (12-14) = -2, (14-14) = 0, (16-14) = 2, (18-14) = 4 - Step 3: Square Deviations
(-4)² = 16, (-2)² = 4, 0² = 0, 2² = 4, 4² = 16 - Step 4: Sum of Squared Deviations
16 + 4 + 0 + 4 + 16 = 40 - Step 5: Calculate Variance
Population: σ² = 40 / 5 = 8
Sample: s² = 40 / (5-1) = 40 / 4 = 10 - Step 6: Calculate Standard Deviation
Population: σ = √8 ≈ 2.83
Sample: s = √10 ≈ 3.16
When to Use Population vs Sample Variance
| Situation | Use Population Variance | Use Sample Variance | Example |
|---|---|---|---|
| Data Scope | Complete population data | Sample from larger population | Census vs survey |
| Denominator | N (actual count) | n-1 (degrees of freedom) | 5 vs 4 for n=5 |
| Purpose | Describe population | Estimate population parameter | Parameter vs statistic |
| Statistical Test | Descriptive statistics | Inferential statistics | z-test vs t-test |
| Result Value | Slightly smaller | Slightly larger (unbiased) | Corrects for sampling error |
Real-World Applications
Quality Control & Manufacturing
- Process control: Monitor production consistency using variance
- Product specifications: Ensure products meet tolerance limits
- Six Sigma: Measure process capability (Cp, Cpk)
- Supplier evaluation: Compare consistency of different suppliers
Finance & Investment
- Risk assessment: Variance measures investment volatility
- Portfolio optimization: Modern Portfolio Theory uses variance-covariance matrix
- Option pricing: Black-Scholes model uses variance (volatility)
- Value at Risk (VaR): Quantify potential losses
Science & Research
- Experimental error: Measure precision of measurements
- Statistical testing: ANOVA, regression analysis use variance
- Data validation: Check consistency of experimental results
- Signal processing: Separate signal from noise
Business & Economics
- Sales forecasting: Measure sales variability
- Inventory management: Analyze demand variability
- Performance evaluation: Compare consistency of sales teams
- Market research: Analyze consumer behavior variability
Healthcare & Medicine
- Clinical trials: Measure treatment effect variability
- Biological measurements: Analyze natural variation
- Diagnostic tests: Assess test reliability
- Epidemiology: Study disease spread patterns
Properties of Variance
| Property | Formula | Explanation | Example |
|---|---|---|---|
| Non-negativity | σ² ≥ 0 | Variance is always positive or zero | All data same → σ² = 0 |
| Additivity | Var(aX + b) = a²Var(X) | Scaling affects variance by square | Double values → 4× variance |
| Linearity | Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y) | For independent: Var(X+Y) = Var(X) + Var(Y) | Independent variables add |
| Constant | Var(c) = 0 | Constant has zero variance | Var(5) = 0 |
| Units | Units² | Variance has squared units | If data in cm, variance in cm² |
Common Variance Values Interpretation
Zero Variance (σ² = 0)
All data points are identical
Example: [5, 5, 5, 5]
Interpretation: No variability
Low Variance
Data points close to mean
Example: Test scores [85, 87, 86, 88]
Interpretation: Consistent results
High Variance
Data points spread out
Example: Test scores [50, 70, 90, 100]
Interpretation: Inconsistent results
Relative Variance
Coefficient of Variation (CV)
CV = σ/μ × 100%
Measures relative variability
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Frequently Asked Questions (FAQs)
Q: What's the difference between variance and standard deviation?
A: Variance is the average of squared deviations from mean (units squared). Standard deviation is the square root of variance (original units). Example: If data is in meters, variance is in m², standard deviation is in meters. Standard deviation is more interpretable.
Q: Why do we use n-1 for sample variance?
A: Using n-1 (Bessel's correction) provides an unbiased estimate of population variance. When using sample mean (which is estimated from data), we lose one degree of freedom. n-1 corrects for this and prevents underestimation of true population variance.
Q: Can variance be negative?
A: No, variance can never be negative. Since it's calculated by squaring deviations, all terms are positive or zero. Zero variance occurs only when all data points are identical.
Q: What does high variance indicate?
A: High variance indicates that data points are spread out over a wider range of values. This could mean: 1) More risk/volatility in finance, 2) Less consistency in manufacturing, 3) More diversity in populations, 4) Less reliable measurements in science.
Q: How do I interpret variance in practical terms?
A: Use standard deviation (square root of variance) for interpretation. For normally distributed data: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ. Example: If test scores have σ=10, most scores (95%) are within ±20 points of mean.
Calculate data variability with Toolivaa's free Variance Calculator, and explore more statistical tools in our Math Calculators collection.