Calculate Pearson correlation coefficient r from paired x and y values, then see the direction, strength, r², covariance-style sums, and clear step-by-step work.
Pearson correlation measures how strongly two numeric variables move together in a straight-line pattern. A value near +1 is strong positive correlation, near -1 is strong negative correlation, and near 0 means little linear relationship.
Enter paired data to calculate r.
How to use the correlation calculator
Paste or type paired values where each row contains one x value and one y value. The first number is treated as x and the second number is treated as y. The calculator then computes Pearson’s r, the standard linear correlation coefficient used in many algebra, statistics, science, business, and research classes.
Use one matched observation per line, such as test hours and exam score, height and weight, advertising spend and sales, or temperature and demand.
The sign tells direction. The size tells strength. r² estimates the share of linear variation explained by the relationship, but it does not prove cause and effect.
Pearson correlation formula
The calculator uses the paired-deviation form of Pearson’s correlation coefficient:
r = Σ((x − x̄)(y − ȳ)) ÷ √[Σ(x − x̄)² · Σ(y − ȳ)²]
Here, x̄ is the mean of the x values and ȳ is the mean of the y values. For each pair, the calculator finds how far x and y are from their means, multiplies those deviations, and compares the combined movement with the spread in each variable.
Worked example
Suppose the paired data are (1,2), (2,3), (3,5), (4,4), (5,6). The mean of x is 3 and the mean of y is 4. The calculator subtracts each mean from each row, multiplies the paired deviations, and divides by the square root of the two deviation-square sums.
| Step | What happens |
|---|---|
| 1 | Find x̄ and ȳ, the two averages. |
| 2 | Compute each x deviation and y deviation from those averages. |
| 3 | Add the cross-products and the squared deviations. |
| 4 | Apply the Pearson r formula and interpret the sign and strength. |
How to interpret a correlation result
Correlation values always fall from -1 to +1. These labels are practical study guides, not universal scientific cutoffs, because different fields use different standards.
| r value | Common interpretation |
|---|---|
| About +0.70 to +1.00 | Strong positive linear relationship. |
| About +0.30 to +0.69 | Weak to moderate positive relationship. |
| Near 0 | Little or no linear relationship. |
| About -0.30 to -0.69 | Weak to moderate negative relationship. |
| About -0.70 to -1.00 | Strong negative linear relationship. |
Correlation does not prove causation
A high correlation can be useful, but it does not prove that one variable causes the other. A third factor, sampling method, time trend, or outlier can create a strong-looking relationship. For schoolwork, reports, or research, describe the data source and inspect a scatter plot before making a conclusion.
Look for outliers, curved patterns, mixed groups, or repeated measurements. Pearson r is designed for linear association.
Say “x and y are positively correlated” instead of “x causes y” unless you have a research design that supports causation.
Common mistakes
- Mixing unmatched x and y values from different rows.
- Using Pearson correlation for a clearly curved relationship without checking a plot.
- Rounding too early before the final r value.
- Reading r² as proof of causation instead of explained linear variation.
- Ignoring a small sample size or an outlier that controls the result.
Correlation calculator FAQs
What is Pearson r?
Pearson r is a number from -1 to +1 that measures the direction and strength of a linear relationship between two numeric variables.
Can r be greater than 1?
No. If a manual calculation gives a value below -1 or above +1, there is usually an arithmetic, pairing, or formula error.
What does a negative correlation mean?
A negative correlation means that larger x values tend to go with smaller y values in a straight-line pattern. It does not automatically mean the relationship is bad or harmful.
Is this the same as covariance?
No. Covariance measures joint movement in the original units. Correlation standardizes that movement, so the result is always between -1 and +1.
How many data pairs do I need?
The calculator can compute r with two or more pairs, but a meaningful interpretation usually needs more data and a sensible sampling method.
Should I use sample or population correlation?
The Pearson r value is the same under the common sample-deviation form used here because the shared scaling cancels in the ratio. For reporting, follow the method required by your class or analysis guide.
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Reviewer
This educational math content is reviewed under the SolverCalc reviewer rule by Prof. Dr. Khalil Mudassar. WordPress adds the author bio automatically, so this page does not include a duplicate author box.
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