Covariance Calculator
Calculate covariance, correlation coefficient, and analyze relationships between variables with step-by-step statistical analysis. Our statistics calculator supports portfolio analysis, covariance matrices, and comprehensive data relationship studies.
Last updated: December 15, 2024
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Enter numbers separated by commas or spaces
Must have same number of values as Dataset X
Statistical Analysis
Covariance:
5.000000
Correlation (r):
1.0000
Relationship:
Positive relationship
Mean X: 3.0000
Var X: 2.5000
Mean Y: 6.0000
Var Y: 10.0000
Analysis:
Covariance of 5.0000 indicates a positive relationship. As X increases, Y tends to increase. The correlation coefficient of 1.0000 provides a standardized measure.
Calculation Steps:
- Data points: n = 5
- Mean X: 3.0000, Mean Y: 6.0000
- Covariance formula: Σ(Xi - X̄)(Yi - Ȳ)/n-1
- Covariance = 5.0000
- Correlation = 1.0000
Covariance Properties:
- • Positive: Variables tend to increase together
- • Negative: One increases while other decreases
- • Zero: No linear relationship
- • Correlation: Standardized covariance (-1 to +1)
Quick Example Result
For datasets X = [1,2,3,4,5] and Y = [2,4,6,8,10]:
Covariance = 2.5000, r = 0.9487
Strong positive relationship - variables increase together
How This Calculator Works
Our covariance calculator applies fundamental statistical principles to analyze relationships between two variables. The calculator uses covariance formulasto measure how variables change together and provides correlation coefficients for standardized interpretation.
Statistical Formulas
Cov(X,Y) = Σ(Xi - X̄)(Yi - Ȳ) / (n-1)Cov(X,Y) = Σ(Xi - X̄)(Yi - Ȳ) / nr = Cov(X,Y) / (σx × σy)These formulas measure how two variables vary together. Sample covariance uses n-1 (Bessel's correction) for unbiased estimation, while population covariance uses n. Correlation standardizes covariance to values between -1 and +1 for easier interpretation.
Shows positive, negative, and zero covariance patterns in data relationships
Statistical Foundation
Covariance is a fundamental measure in statistics that quantifies the joint variability of two random variables. Unlike variance, which measures how a single variable varies from its mean, covariance measures how two variables vary together. This concept is essential in multivariate statistics, portfolio theory, and data analysis.
- Positive covariance indicates variables tend to increase or decrease together
- Negative covariance suggests an inverse relationship between variables
- Zero covariance implies no linear relationship (but nonlinear relationships may exist)
- Covariance magnitude depends on variable scales, making correlation more interpretable
Sources & References
- Introduction to Mathematical Statistics - Robert V. Hogg, Joseph McKean, Allen T. Craig (8th Edition)Comprehensive treatment of covariance and correlation theory
- American Statistical Association - Statistical Education GuidelinesProfessional standards for teaching covariance and correlation concepts
- NIST Engineering Statistics Handbook - Exploratory Data AnalysisPractical applications and computational methods for covariance
Need help with other statistical calculations? Check out our correlation calculator and variance calculator.
Get Custom Calculator for Your PlatformExample Analysis
Stock Returns (%):
- Stock A: [5, 3, 7, 2, 6]
- Stock B: [4, 2, 8, 1, 5]
- Analysis: Portfolio risk assessment
Statistical Results:
- Mean A: 4.6%, Mean B: 4.0%
- Covariance: 4.55 (positive relationship)
- Correlation: r = 0.89 (strong positive)
- Portfolio diversification: Limited
Result: High positive covariance indicates limited diversification benefit
The strong positive correlation (0.89) between stocks A and B means they tend to move together, providing little risk reduction through diversification. Investors might seek assets with lower or negative covariance for better portfolio balance.
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