Covariance Calculator
Calculate sample and population covariance between two variables with detailed statistical analysis
Calculate Covariance
Enter Data Points (X, Y)
Covariance Results
Interpretation: Very weak relationship between variables
Relationship Strength: Very Weak
Sample Size: 5 data points
Covariance Interpretation
Example: Stock Price Analysis
Investment Portfolio Analysis
Scenario: Analyzing relationship between two stock prices
Stock A (Cool Places): [12.76, 12.35, 12.43, 12.70, 13.09]
Stock B (Star Dust): [7.06, 6.81, 6.88, 6.98, 7.35]
Calculation Results
Stock A Mean: 12.666, Stock B Mean: 7.016
Sample Covariance: 0.0604
Population Covariance: 0.0483
Interpretation: Positive covariance indicates stocks tend to move together
Investment Decision
For portfolio diversification, look for stocks with covariance close to zero or negative, as they provide better risk distribution.
Covariance vs Other Measures
Covariance
Measures joint variability
Units: product of X and Y units
Correlation
Normalized covariance
Range: -1 to +1
Variance
Covariance of X with itself
Var(X) = Cov(X,X)
Covariance Tips
Positive covariance: variables increase together
Negative covariance: variables move oppositely
Zero covariance: no linear relationship
Sample covariance divides by (n-1)
Population covariance divides by n
Use correlation for scale-free comparison
Understanding Covariance
What is Covariance?
Covariance measures how two variables change together. It indicates whether increases in one variable tend to be associated with increases (positive covariance) or decreases (negative covariance) in another variable.
Sample vs Population Covariance
- •Sample covariance: Uses n-1 in denominator (Bessel's correction)
- •Population covariance: Uses n in denominator
- •Sample covariance provides unbiased estimate of population covariance
Formulas
Sample Covariance
Cov(X,Y) = Σ(xi - x̄)(yi - ȳ) / (n-1)
Population Covariance
Cov(X,Y) = Σ(xi - x̄)(yi - ȳ) / n
Correlation Coefficient
r = Cov(X,Y) / (σx × σy)
Applications
Finance
Portfolio diversification, risk management, asset correlation analysis
Research
Studying relationships between variables, hypothesis testing, data analysis
Machine Learning
Feature selection, dimensionality reduction, pattern recognition