Enter paired values
Provide X and Y values in matching order. Separate numbers with commas, spaces, or new lines.
Example data table
| Observation | X value | Y value | Context |
|---|---|---|---|
| 1 | 10 | 15 | Lower feature score with lower conversion quality. |
| 2 | 12 | 18 | Small lift in both measured variables. |
| 3 | 14 | 19 | Moderate improvement in the second variable. |
| 4 | 16 | 23 | Stronger paired movement appears clearly. |
| 5 | 18 | 25 | Consistent positive association remains visible. |
| 6 | 20 | 28 | Highest values reinforce the linear pattern. |
Formula used
The calculator uses the Pearson product moment correlation coefficient for paired observations. It compares how far each X and Y value sits from its own mean, then scales that shared movement by both standard deviations.
Pearson correlation:
r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √(Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²)
Sample covariance:
Cov(X,Y) = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / (n − 1)
Linear trend line:
Slope = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / Σ(xᵢ − x̄)²
Intercept = ȳ − Slope × x̄
For significance testing, the calculator converts r into a t statistic using t = r × √((n − 2) / (1 − r²)). It also builds a confidence interval for r using Fisher’s z transformation when enough observations are available.
How to use this calculator
- Enter one list of X values and one list of Y values.
- Keep both lists in the same paired order.
- Choose decimal precision, confidence level, and significance level.
- Select whether invalid pairs should stop or be skipped.
- Press Calculate correlation to show the result above the form.
- Review r, R², covariance, p value, and confidence interval.
- Use the CSV or PDF buttons to export the output.
Frequently asked questions
1. What does the correlation coefficient show?
It shows the direction and strength of a linear relationship between two paired variables. Values near 1 or −1 indicate stronger linear patterns, while values near 0 suggest little linear association.
2. Can a strong correlation prove causation?
No. Correlation only measures association within the data you entered. Outside factors, hidden variables, or reverse effects may explain the pattern, so causal claims need stronger study design.
3. Why does the calculator need paired data?
Each X value must align with the correct Y value from the same observation. If the order is mixed, the relationship estimate becomes misleading because the matched structure is lost.
4. What does R² mean here?
R² is the square of Pearson’s r. It estimates how much variation in one variable is explained by a simple linear relationship with the other variable.
5. Why can’t I calculate correlation with zero variance?
If every X value or every Y value is identical, the denominator of the correlation formula becomes zero. Without variability, the relative movement between variables cannot be standardized.
6. What is the p value testing?
The p value tests whether the true population correlation could reasonably be zero. A small p value suggests the observed linear relationship is unlikely to be explained by random sampling alone.
7. When should I use Spearman instead of Pearson?
Use Spearman correlation when the relationship is monotonic but not linear, when ranks matter more than raw values, or when outliers strongly distort Pearson’s correlation.
8. Does this tool handle missing or invalid values?
Yes. You can choose to stop the calculation immediately or skip invalid matched pairs. Skipping works best when a few entries are missing but the remaining pairs stay trustworthy.