What Is Pearson Correlation?

The Pearson product-moment correlation coefficient, denoted $r$, quantifies the strength and direction of the linear relationship between two continuous variables. It answers the question: "As one variable increases, does the other tend to increase (positive), decrease (negative), or show no consistent pattern?"

The value of $r$ ranges from $-1$ to $+1$:

It is important to remember that $r$ measures linear relationships only. Two variables can have a strong curvilinear relationship yet produce an $r$ near zero.

When to Use It

Use the Pearson correlation when:

• You have two continuous variables measured on interval or ratio scales (e.g., test scores, age, income, reaction time).
• You want to describe the direction and strength of a relationship rather than predict one variable from another (for prediction, use regression).
• Your data appear to follow a roughly linear pattern when plotted in a scatter plot.

If one or both variables are ordinal (e.g., Likert-scale items, ranked data), consider Spearman's rank correlation ($r_s$) instead.

Assumptions

Before computing $r$, verify these assumptions:

1. Level of measurement. Both variables must be continuous (interval or ratio).
2. Linearity. The relationship between the two variables should be linear. Always inspect a scatter plot first.
3. Bivariate normality. The pair of variables should be approximately normally distributed. With large samples ($n > 30$), the test is robust to moderate violations.
4. No significant outliers. Outliers can dramatically inflate or deflate $r$. A single extreme point can turn a weak correlation into a strong one (or vice versa).
5. Homoscedasticity. The spread of one variable should be roughly constant across levels of the other variable. A "funnel" shape in the scatter plot signals a violation.

Formula

The Pearson correlation coefficient is calculated as:

$$r = \frac{\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_{i=1}^{n}(X_i - \bar{X})^2 \cdot \sum_{i=1}^{n}(Y_i - \bar{Y})^2}}$$

Where:

• $X_i$ and $Y_i$ are individual data points
• $\bar{X}$ and $\bar{Y}$ are the sample means of $X$ and $Y$
• $n$ is the number of paired observations

The numerator is the sum of cross-products of deviations, which captures how $X$ and $Y$ co-vary. The denominator standardizes this quantity by the total variability in each variable.

Coefficient of Determination

The square of the correlation, $r^2$, is called the coefficient of determination. It tells you the proportion of variance in one variable that is explained by (shared with) the other.

$$r^2 = \text{proportion of shared variance}$$

For example, if $r = .60$, then $r^2 = .36$, meaning 36% of the variance in $Y$ is accounted for by its linear relationship with $X$.

Testing Significance

To test whether $r$ is significantly different from zero, compute the $t$-statistic:

$$t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$$

This follows a $t$-distribution with $df = n - 2$. If $|t|$ exceeds the critical value, you reject $H_0: \rho = 0$.

Worked Example

Scenario: A health psychologist wants to know whether the number of hours of sleep per night is associated with self-reported stress scores (0--50 scale) among university students. She collects data from $n = 8$ participants.

Step 1: Compute the means.

$$\bar{X} = \frac{5+6+7+6+8+7+9+8}{8} = \frac{56}{8} = 7.0$$

$$\bar{Y} = \frac{38+32+25+30+20+28+15+22}{8} = \frac{210}{8} = 26.25$$

Step 2: Compute deviations and products.

Step 3: Sum the columns.

$$\sum(X_i - \bar{X})(Y_i - \bar{Y}) = -66.00$$

$$\sum(X_i - \bar{X})^2 = 12.00$$

$$\sum(Y_i - \bar{Y})^2 = 373.50$$

Step 4: Compute $r$.

$$r = \frac{-66.00}{\sqrt{12.00 \times 373.50}} = \frac{-66.00}{\sqrt{4482.00}} = \frac{-66.00}{66.95} = -0.986$$

Step 5: Test significance.

$$t = \frac{-0.986\sqrt{8 - 2}}{\sqrt{1 - 0.972}} = \frac{-0.986 \times 2.449}{\sqrt{0.028}} = \frac{-2.415}{0.167} = -14.46$$

With $df = 6$ and a critical $t$ of $\pm 2.447$ at $\alpha = .05$ (two-tailed), $|t| = 14.46$ far exceeds the critical value, so $p < .001$.

Step 6: Coefficient of determination.

$$r^2 = (-0.986)^2 = 0.972$$

About 97% of the variance in stress scores is explained by hours of sleep in this sample.

Interpretation

The results show a strong negative correlation between hours of sleep and stress scores, $r = -.99$. As sleep increases, stress decreases in a nearly perfect linear pattern. The coefficient of determination ($r^2 = .97$) indicates that 97% of the variability in stress scores can be accounted for by the linear relationship with sleep hours.

Keep in mind:

• Correlation does not imply causation. We cannot conclude that more sleep causes lower stress. A third variable (e.g., workload) might drive both.
• Strength benchmarks are field-dependent. In psychology, $r = .30$ can be practically meaningful; in physics, $r = .90$ might be considered weak.
• $r$ only captures linear relationships. If the scatter plot shows a curve, consider polynomial regression or Spearman's correlation.

Common Mistakes

1. Assuming causation. Correlation quantifies association, not cause and effect. Always consider confounding variables and the direction-of-causality problem.
2. Ignoring outliers. A single extreme data point can flip the sign or magnitude of $r$. Always inspect scatter plots and consider running analyses with and without potential outliers.
3. Restricting the range. If you only sample from a narrow range of one variable (e.g., only high-achieving students), $r$ will be artificially attenuated. This is called range restriction.
4. Confusing $r$ with $r^2$. Reporting $r = .50$ as "50% of variance explained" is wrong. The correct figure is $r^2 = .25$, or 25%.
5. Using Pearson's $r$ with non-linear data. If the scatter plot shows a curved pattern, $r$ will underestimate the true strength of the relationship. Use a non-linear method or transform the data.
6. Using Pearson's $r$ with ordinal data. Likert-scale items (e.g., 1--5 agreement ratings) are ordinal. Use Spearman's $r_s$ or polychoric correlation instead.

How to Run It