Grades Geometry & Functions StatisticsCorrelation & Linear Regression

Correlation & Linear Regression

📊 Statistics

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Correlation & Linear Regression

Quantify relationships between variables and fit lines to data.

Linear regression finds the line of best fit (least-squares line) that minimizes the sum of squared vertical distances from data points to the line. The correlation coefficient r measures the strength and direction of the linear relationship.

r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i-\bar{x})^2 \cdot \sum(y_i-\bar{y})^2}}

Interpreting r

r = 1: perfect positive linear relationshipr = −1: perfect negative linear relationshipr = 0: no linear relationship|r| > 0.8: strong correlation0.5 < |r| < 0.8: moderate correlation

\hat{y} = a + bx \quad \text{where } b = r\cdot\frac{s_y}{s_x},\; a = \bar{y} - b\bar{x}

⚠️

Correlation does not imply causation. A high |r| value means the variables are linearly related, not that one causes the other.

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Remember This!

Residuals = observed y − predicted ŷ. If the residual plot shows a random scatter (no pattern), the linear model is appropriate. A curved pattern suggests a non-linear model is better.

✏️ Try It!

A linear regression gives ŷ = 2.5x + 10. What does the slope 2.5 mean in context?

Practice with CalcVerse

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