Correlation Calculator
Calculate the Pearson correlation coefficient (r) between two datasets. Determine the strength and direction of the linear relationship, R², and statistical significance. See also our Linear Regression Calculator, R-Squared Calculator, and Coefficient of Variation Calculator.
How to Use the Correlation Calculator
The Pearson correlation coefficient measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation). This calculator computes r, R², an approximate p-value, and interprets the strength of the relationship.
Enter your x values and y values as comma-separated numbers. Both datasets must have the same number of values, and at least 3 pairs are required. The calculator pairs the first x with the first y, the second x with the second y, and so on. Click Calculate to see the correlation coefficient and its interpretation.
The p-value tests whether the correlation is statistically significantly different from zero. A small p-value (typically < 0.05) suggests the correlation is unlikely to have occurred by chance. However, statistical significance does not imply practical significance — with large samples, even tiny correlations can be statistically significant. Always consider the magnitude of r alongside the p-value.
The strength interpretation follows standard guidelines: |r| above 0.7 is considered strong, 0.5-0.7 is moderate, 0.3-0.5 is weak, and below 0.3 indicates little to no linear relationship. Remember that Pearson's r only measures linear relationships — two variables can have a strong non-linear association (quadratic, exponential) while showing a near-zero Pearson correlation.
Formula
Pearson Correlation Coefficient:
r = (nΣxy - ΣxΣy) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]
Alternative form (using means):
r = Σ(xᵢ - x̄)(yᵢ - ȳ) / √[Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²]
Coefficient of Determination:
R² = r²
t-statistic for significance:
t = r√(n-2) / √(1-r²)
Degrees of freedom:
df = n - 2
Confidence interval for r (Fisher z-transform):
z = 0.5 × ln((1+r)/(1-r)), SE_z = 1/√(n-3)
Example Calculation
X: 1, 2, 3, 4, 5
Y: 2, 4, 5, 4, 5
n = 5, Σx = 15, Σy = 20, Σxy = 67, Σx² = 55, Σy² = 86
r = (5×67 - 15×20) / √[(5×55 - 225)(5×86 - 400)]
r = (335 - 300) / √[(275-225)(430-400)]
r = 35 / √(50×30) = 35 / 38.73 = 0.9037
R² = 0.8167 (81.67% variance explained)
Interpretation: Strong positive correlation
Reference Table
| |r| Range | Strength | Meaning |
|---|---|---|
| 0.90 to 1.00 | Very Strong | Nearly perfect linear relationship |
| 0.70 to 0.89 | Strong | Clear linear trend with some scatter |
| 0.50 to 0.69 | Moderate | Noticeable trend but considerable scatter |
| 0.30 to 0.49 | Weak | Slight trend, much variability |
| 0.00 to 0.29 | Very Weak | No meaningful linear relationship |
| -0.30 to -0.49 | Weak Negative | Slight inverse trend |
| -0.70 to -0.89 | Strong Negative | Clear inverse linear trend |
| -0.90 to -1.00 | Very Strong Negative | Nearly perfect inverse relationship |
Frequently Asked Questions
What is the Pearson correlation coefficient?
The Pearson correlation coefficient (r) quantifies the linear relationship between two continuous variables. It ranges from -1 to +1, where +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. It measures only linear associations — two variables can have a strong non-linear relationship with r near zero. It is the most commonly used measure of correlation in statistics.
Does correlation imply causation?
No. Correlation measures association, not causation. Two variables can be correlated because one causes the other, because they share a common cause (confounding variable), or purely by coincidence (spurious correlation). Establishing causation requires controlled experiments, temporal precedence, and ruling out alternative explanations. For example, ice cream sales and drowning rates are correlated, but both are caused by hot weather — ice cream does not cause drowning.
When should I use Spearman instead of Pearson?
Use Spearman's rank correlation when: (1) the relationship is monotonic but not linear, (2) data contains outliers that would distort Pearson's r, (3) variables are ordinal (ranked) rather than continuous, or (4) the data is not normally distributed. Spearman's rho measures the strength of the monotonic relationship by correlating the ranks of the data rather than the raw values. It is more robust to outliers and non-normality.
What sample size do I need for reliable correlation?
For detecting moderate correlations (r ≈ 0.3) with 80% power at α = 0.05, you need approximately 85 observations. For strong correlations (r ≈ 0.5), about 30 observations suffice. With fewer than 10 observations, correlation estimates are highly unstable and confidence intervals are very wide. As a practical guideline, aim for at least 30 pairs for exploratory analysis and more for confirmatory studies.
How do I interpret R-squared from correlation?
R² = r² represents the proportion of variance in one variable that is explained by the linear relationship with the other. If r = 0.7, then R² = 0.49, meaning 49% of the variance in y is accounted for by x (and vice versa). The remaining 51% is due to other factors or random variation. R² is always between 0 and 1 regardless of the sign of r, making it useful for comparing the explanatory power of different relationships.
Can correlation be used with categorical data?
Pearson's r requires continuous (interval or ratio) data. For categorical data, use alternative measures: point-biserial correlation for one dichotomous and one continuous variable, phi coefficient for two dichotomous variables, or Cramér's V for nominal variables with more than two categories. For ordinal data, use Spearman's rho or Kendall's tau. Using Pearson's r on categorical data produces meaningless results.