Enter a dataset to calculate mean, variance, standard deviation, and coefficient of variation
Enter Data (separate values with commas)
Results
Step-by-Step Calculation
Variance & Standard Deviation Formulas
Mean：x̄ = (x₁+x₂+...+xn)/n
Population Variance：σ² = Σ(xᵢ-x̄)²/n
Sample Variance：s² = Σ(xᵢ-x̄)²/(n-1)
Standard Deviation：σ = √σ² or s = √s²
Variance and standard deviation are key measures of data dispersion. Standard deviation uses the same unit as raw data for better interpretability.
⚠Population variance uses denominator n, sample variance uses n-1. Enter at least 2 values. Results are rounded to 6 decimal places.
What are Variance and Standard Deviation?
Variance and standard deviation are core descriptive statistics measuring data spread. Variance is the mean of squared deviations from the average; standard deviation is its square root. They quantify data fluctuation and are widely used in data analysis, quality control, and finance.
Dispersion
Larger variance/SD = more spread data; smaller values = more concentrated data. SD = 0 means all values are identical.
Mean Reference
Calculated from squared deviations from the mean. The sum of all deviations from the mean is always zero.
Units
Variance uses squared units; standard deviation matches original data units, making it more practical for interpretation.
Coefficient of Variation
CV=σ/μ×100%, used to compare dispersion across datasets with different units or widely different means.
💡 Example: Data: 85,90,78,92,88. Mean = (85+90+78+92+88)/5 = 86.6. Squared deviations sum = 119.2. Population variance = 119.2/5 = 23.84. Standard deviation = √23.84 ≈ 4.88.
Applications
Data AnalysisQuality ManagementFinancial RiskResearch & ExperimentsMachine Learning
Frequently Asked Questions
What is the difference between variance and standard deviation?▼
Variance is the average of squared differences from the Mean, measuring data spread. Standard Deviation is the positive square root of variance, with the same unit as original data, making it more intuitive for describing dispersion.
What is the difference between population and sample variance?▼
Population variance uses all data (denominator = n). Sample variance estimates a population (denominator = n-1, Bessel’s correction) to remove bias. The difference is negligible for large sample sizes.
What is the Coefficient of Variation (CV)?▼
CV = (Standard Deviation / Mean) × 100%. It measures relative dispersion, eliminating unit and mean scale effects, ideal for comparing datasets with different units or large mean differences.
What are practical uses of standard deviation?▼
Standard deviation is widely used in: quality control (6σ), financial risk (volatility), test score analysis, experimental error measurement, data standardization for machine learning, and statistical process control.
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