Enter Paired Values
Use commas, spaces, semicolons, tabs, or new lines. Keep both series aligned so each X matches one Y.
Example Data Table
This sample shows a clear positive relationship between both variables.
| Observation | X | Y |
|---|---|---|
| 1 | 12 | 30 |
| 2 | 15 | 34 |
| 3 | 18 | 40 |
| 4 | 20 | 43 |
| 5 | 22 | 47 |
| 6 | 26 | 53 |
Formulas Used
Mean(X) = ΣX / nMean(Y) = ΣY / n
Cov(X,Y) = Σ[(Xi - X̄)(Yi - Ȳ)] / (n - 1)
Cov(X,Y) = Σ[(Xi - μx)(Yi - μy)] / n
r = Σ[(Xi - X̄)(Yi - Ȳ)] / √(Σ(Xi - X̄)² × Σ(Yi - Ȳ)²)
Slope = Σ[(Xi - X̄)(Yi - Ȳ)] / Σ(Xi - X̄)²Intercept = Ȳ - Slope × X̄
Covariance keeps the data units, so its magnitude depends on scale. Correlation is standardized, making comparison across datasets easier.
How to Use This Calculator
- Enter one list for X values and one matching list for Y values.
- Choose automatic parsing or set the exact delimiter manually.
- Select sample covariance for sample data or population covariance for full datasets.
- Set decimal precision and optional chart labels.
- Press Calculate Now to show results above the form.
- Review the scatter plot, regression line, covariance, correlation, and summary metrics.
- Use the export buttons to download the CSV or PDF report.
Frequently Asked Questions
1) What does covariance measure?
Covariance measures whether two variables tend to move together. Positive values suggest they rise together, negative values suggest opposite movement, and values near zero suggest weak linear co-movement.
2) What does correlation measure?
Correlation measures the strength and direction of a linear relationship on a fixed scale from -1 to 1. It removes units, making comparisons across datasets easier.
3) When should I use sample covariance?
Use sample covariance when your values come from a subset of a larger population. It divides by n - 1, which reduces bias in estimating population variability.
4) Can correlation miss nonlinear relationships?
Yes. Pearson correlation focuses on linear patterns. A curved but strong relationship can still produce a modest correlation value, so always inspect the scatter plot too.
5) Why can correlation be undefined?
Correlation becomes undefined when one variable has zero variance. If every X or every Y value is identical, there is no spread to standardize.
6) Why must the values be paired?
Each X value must match the correct Y value from the same observation. Misaligned pairs distort deviations, covariance, correlation, and the fitted regression line.
7) Does negative covariance mean a bad result?
No. Negative covariance only means one variable tends to rise when the other falls. Whether that is desirable depends entirely on your statistical context.
8) Can I export both data and results?
Yes. The page provides CSV and PDF export options after calculation, including summary metrics and the paired dataset used for analysis.