Pearson Correlation & Covariance Calculator
Paste matched X and Y observations to measure linear association and inspect covariance on the original measurement scales.
Correlation standardizes association; covariance keeps the original scale
Pearson r divides the paired cross-deviation by both variables’ spread, producing a unitless measure from −1 to 1. Covariance uses the same paired deviations without that standardization, so its magnitude depends on the units of X and Y.
Lynara reports both sample covariance and population covariance because the denominator depends on whether the entered pairs are treated as a sample or the complete population of interest. Neither statistic establishes a causal relationship.
Privacy: calculator inputs are processed in your browser; Lynara does not need to upload these values to calculate the result.
Formula
r = cross-deviation sum ÷ √(SSx × SSy); covariance = cross-deviation sum ÷ n or (n − 1)
Perfect correlation can reverse sign without changing either set of values
- Pair X = [1, 2, 3, 4] with Y = [2, 4, 6, 8]. Each increase of 1 in X comes with an increase of 2 in Y. The Pearson coefficient is r = +1, indicating a perfect positive linear relationship in this invented data.
- Keep X fixed but reverse the order of Y to [8, 6, 4, 2]. The same four X and four Y values are present, but the pairs now move in opposite directions, so Pearson r = −1. Matching the rows matters: the calculator treats each row as an observed X–Y pair.
- To see why the coefficient alone is not the full story, imagine repeating the original four pairs many times: the computed correlation stays +1, but repetition does not create a causal experiment or a representative sample.
What the result tells you
Correlation measures how paired observations vary together, not how similar their separate distributions are. A sign reversal with the same individual values demonstrates why preserving each paired measurement is essential.
Before you use the result
Pearson r does not imply causation, and it can be distorted by outliers or non-linear relationships. An r of +1 or −1 on four invented points is not evidence about a population; inspect a scatterplot and sampling method before interpreting real data.
References for this method
Useful for
- Statistics homework
- Paired-data correlation
- Sample and population covariance
- Checking linear association
Questions
What is the difference between correlation and covariance?
Covariance keeps the variables’ units and scale, while Pearson r standardizes the relationship to a unitless value from −1 to 1.
Does r = 1 prove one variable causes the other?
No. Perfect linear association does not establish causation.