Correlation Coefficient Calculator

Which correlation coefficient does this calculate, and what is the exact formula?

Pearson’s product-moment coefficient: r = Σ(x − x̄)(y − ȳ) ÷ √[Σ(x − x̄)² · Σ(y − ȳ)²], using the means of the five X values and five Y values. The n − 1 terms that appear in covariance and standard deviation cancel, so the same r results whether you think of the data as a sample or a population. It measures straight-line association only.

How many data pairs can I enter?

Exactly five. There are five X boxes and five Y boxes, and an empty box is turned into 0 by the form before the calculation, so a missing value is treated as a real zero rather than skipped. You cannot calculate r for three or four pairs, and for larger datasets use a spreadsheet function such as CORREL.

Does the calculator tell me whether the correlation is statistically significant?

No, it reports r, R² and a descriptive label only, with no p-value. With five pairs there are 3 degrees of freedom, and a two-tailed test at the 5% level needs |r| of about 0.878 or more. The example data 1–5 against 2, 4, 5, 4, 5 gives r = 0.7746, which looks "strong" in the label but corresponds to p ≈ 0.124 (t = 2.1213 on 3 df), so it is not significant.

What do the words Very strong, Strong, Moderate and Weak mean?

They are fixed cut-offs in this tool’s code, applied to |r|: 0.9 or above is Very strong, 0.7 to below 0.9 Strong, 0.5 to below 0.7 Moderate, 0.3 to below 0.5 Weak, and anything lower Very weak. These are common rules of thumb, not universal standards. What counts as a strong correlation differs between physics, psychology and marketing data, so use the number itself in reports.

Why do I get an error saying "No variation in data"?

Either all five X values are identical or all five Y values are identical. Then one of the sums of squares in the denominator is zero and r is undefined, because a constant cannot rise or fall with anything. Change at least one value in each column.

Can r be zero when the variables are clearly related?

Yes. For X = −2, −1, 0, 1, 2 and Y = 4, 1, 0, 1, 4, Y is fully determined by X (it is X squared), yet the calculator returns r = 0 and "No linear correlation", because the upward and downward halves cancel. Always plot the points before relying on r.

How much can a single outlier change the result?

A great deal with so few points. With X = 1–5 and Y = 1, 2, 3, 4, 5, r is exactly 1.0000. Changing only the last Y to 20 drops r to 0.8000, even though the first four points still sit on a perfect line. Pearson r uses squared distances, so extreme values dominate it.

What does R² (explained) show, and does a high r prove one variable affects the other?

R² is simply r squared, shown as a percentage: for r = −0.9791 it is 95.87%, the share of the variation in Y that a straight line on X accounts for. It says nothing about cause. Two series can move together because of a third factor, a shared time trend, or chance, which is especially easy with five points.