Calculation Results
Covariance Matrix
Correlation Matrix
Correlation removes measurement scale, making relationship strength easier to compare.
Supporting Matrices
Pairwise Observation Counts
Each entry shows the number of valid observation pairs used.
Centered Sum-of-Products Matrix
Each entry is the covariance numerator before division by its selected denominator.
Descriptive Statistics
Eigenvalue and PCA Analysis
Interpretation and Machine Learning Insights
Calculation Steps and Report
Merged text reportDataset Input
Formula Used
Cov(X, Y) = Σ[(xᵢ − x̄)(yᵢ − ȳ)] ÷ (n − 1)
Cov(X, Y) = Σ[(xᵢ − μₓ)(yᵢ − μᵧ)] ÷ n
Σ = (X − X̄)ᵀ(X − X̄) ÷ (n − 1)
Covw(X, Y) = Σ[wᵢ(xᵢ − μwx)(yᵢ − μwy)] ÷ Σwᵢ
How to Use
- Paste a numeric dataset or upload a CSV file.
- Choose the delimiter and dataset orientation.
- Select sample, population, or weighted covariance.
- Choose missing-value and preprocessing options.
- Detect variables and select the required features.
- Set precision, matrix view, and display preferences.
- Calculate and review matrices, diagnostics, and PCA results.
- Copy, print, or download the generated outputs.
Example Dataset
This housing example compares size, bedrooms, property age, and price.
| Size | Bedrooms | Age | Price |
|---|---|---|---|
| 1200 | 2 | 18 | 210000 |
| 1500 | 3 | 12 | 275000 |
| 1700 | 3 | 8 | 315000 |
| 2000 | 4 | 5 | 390000 |
| 2300 | 4 | 3 | 455000 |
Frequently Asked Questions
What does a covariance matrix show?
It shows each variable variance and every pairwise covariance.
Why are diagonal entries variances?
A variable’s covariance with itself equals its variance.
Why is the covariance matrix symmetric?
Cov(X, Y) equals Cov(Y, X), producing mirrored entries.
Should I use sample or population covariance?
Use sample covariance for sampled data and population covariance for complete populations.
How is covariance different from correlation?
Correlation standardizes covariance, keeping values between negative one and positive one.
Why does feature scaling matter?
Covariance magnitude changes with units, so large-scale features can dominate.
How is covariance used in PCA?
PCA decomposes the covariance matrix into eigenvalues and eigenvectors.
What causes a singular covariance matrix?
Duplicate, constant, or linearly dependent features commonly cause singularity.
What does pairwise deletion do?
It uses all available observation pairs for each covariance entry.