Use this Pearson correlation coefficient calculator to measure the linear relationship between two continuous variables. Enter your raw data, upload a CSV or Excel sheet, and instantly see Pearson r, r², p-value, t-statistic, confidence intervals, scatter plot diagnostics, and a full step-by-step solution that is suitable for teaching, assignments, and quick statistical checks.
Best for continuous variables with a linear relationship.
Pearson r = 0.9991 based on 8 valid pairs, p = 0.0000.
Average the X values and the Y values before measuring joint movement.
Subtract the mean from every X and Y value to get centered deviations.
Multiply each pair of deviations and add them up.
Compute the denominator from the independent spread of X and Y.
The covariance term is normalized by both standard-deviation components.
Enter paired X and Y values manually, paste directly from Excel, or import a CSV or Excel file.
Check that both variables are continuous and that a linear relationship is the right assumption for your data.
Read the r value, r², p-value, t-statistic, degrees of freedom, and confidence intervals from the result cards.
Review the scatter plot, residual plot, interpretation card, and worked solution before drawing a conclusion.
The Pearson correlation coefficient is the standard statistic for measuring the strength and direction of a linear relationship between two continuous variables. When people refer to Pearson r, they usually mean the sample estimate of how closely paired observations move together around a straight-line trend. The Pearson correlation coefficientranges 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.
The method is named after Karl Pearson, who formalized the coefficient in modern statistics. A Pearson correlation coefficient is most appropriate when both variables are measured on continuous scales, the pattern is approximately linear, and the data are not being dominated by a few influential outliers. For formal inference, analysts often also check whether the variables are roughly normally distributed, especially in smaller samples.
In practice, Pearson r is useful because it answers several questions at once. It tells you whether the relationship is positive or negative, how strong the relationship is, and whether the observed pattern is statistically distinguishable from zero using a significance test. It also connects naturally to , which gives the proportion of shared variance explained by the linear model. That is why Pearson correlation appears so often in psychology, business analysis, education, health research, and engineering dashboards.
| Symbol | Meaning |
|---|---|
| Pearson correlation coefficient | |
| The ith paired observation | |
| Sample means of X and Y | |
| Number of valid paired observations | |
| Test statistic for significance testing | |
| Degrees of freedom, equal to n minus 2 | |
| Fisher z transformed correlation | |
| Confidence interval for the population correlation |
The first formula converts raw paired data into a standardized covariance. The numerator captures whether X and Y tend to move together, while the denominator scales that movement by the variability present in each variable. That is why Pearson r always stays between -1 and +1.
The t statistic transforms the sample correlation into a significance test under the null hypothesis that the population correlation is zero. Fisher's z transform is then used to build confidence intervals because the sampling distribution of r becomes more symmetric after transformation.
Compute the mean of X and the mean of Y so each variable has a central reference point.
Subtract each mean from every observation to get paired deviations from the center.
Multiply the paired deviations and sum them to create the covariance numerator.
Square the deviations within X and Y, sum them, and multiply the two totals inside the denominator.
Divide the numerator by the denominator to get Pearson r, then test significance with the t formula.
The interactive Step-by-Step module above follows this same logic with your actual numbers. It is useful when you need to verify a homework solution, explain a result in class, or show a client how the coefficient was obtained from the raw data rather than from a black-box tool.
Start with the sign of r. Positive values mean the variables rise together, while negative values mean one tends to fall as the other rises. Then look at the magnitude. A larger absolute value indicates a stronger linear relationship. After that, use the p-value and confidence interval to judge uncertainty. Finally, inspect the scatter plot and residual plot to make sure the linear model is a sensible summary of the data.
Pearson is ideal when the relationship is linear and both variables are continuous. Spearman is a safer choice for ordinal or clearly non-normal data because it works on ranks. Kendall is even more robust in small samples or datasets with many tied values. A practical workflow is to calculate Pearson first, then compare against Spearman or Kendall if the scatter plot looks curved, the residuals look unstable, or outliers appear influential.
| Pearson r | Spearman ρ | Kendall τ | |
|---|---|---|---|
| Data type | Continuous | Ordinal or continuous | Ordinal or continuous |
| Distribution requirement | Approximately normal for formal inference | No strict normality requirement | No strict normality requirement |
| Outlier sensitivity | Sensitive | More robust | Most robust |
| Typical sample size | Usually n ≥ 10 | Usually n ≥ 10 | Small samples handled well |
| Link | Current page | /spearman-correlation/ | /kendall-correlation/ |
A classic positive linear relationship with mild natural variation.
This type of dataset is useful when both variables are measured numerically and you want to quantify a straight-line association rather than only a ranked pattern.
Stronger temperatures generally push sales up, with one cooler-day outlier.
This type of dataset is useful when both variables are measured numerically and you want to quantify a straight-line association rather than only a ranked pattern.
Useful for step-by-step teaching because the pattern is easy to inspect by hand.
This type of dataset is useful when both variables are measured numerically and you want to quantify a straight-line association rather than only a ranked pattern.
A good Pearson correlation coefficient depends on the field. In social sciences, values above 0.5 are often treated as strong. In physics or engineering, values above 0.9 may be expected. As a practical guide, absolute values below 0.3 are usually weak, 0.3 to 0.7 are moderate, and values above 0.7 are strong.
First find the mean of X and Y. Then subtract each mean from its values to get deviations. Multiply each paired deviation and sum the products. Compute the square root of the product of each variable's sum of squared deviations. Finally, divide the covariance numerator by that denominator.
A Pearson r of 0.7 indicates a strong positive linear relationship. About 49% of the variance in Y is explained by X because r squared is 0.49. With a moderate sample size, it is often statistically significant.
r = 0.5 is usually considered a moderate correlation. It means roughly 25% of the variance in one variable is explained by the other. Whether that is strong enough depends on the research setting and the amount of noise normally present in the data.
Pearson correlation measures linear association between two continuous variables and is most appropriate when the relationship is roughly linear. Spearman correlation measures monotonic association using ranked values, so it is more robust to outliers and non-normal data.
Convert r into a t statistic using t = r multiplied by the square root of n minus 2, divided by the square root of 1 minus r squared. Then use the t distribution with n minus 2 degrees of freedom to get the two-tailed p-value. The calculator does this automatically.
Yes. A negative Pearson correlation means the two variables tend to move in opposite directions. As one variable increases, the other tends to decrease. The closer the value is to negative one, the stronger that inverse linear relationship is.
A minimum of about 10 observations is often recommended for a rough estimate, but 30 or more is better for stable inference. Very small samples can produce unstable r values and wide confidence intervals, so significance tests should be interpreted carefully.