AP Precalculus: Bivariate Statistics Formulas & Concepts

1. Outliers in Scatter Plots

• Outlier: point that lies far from the general trend
• May affect correlation/regression results

2. Correlation Coefficient (\( r \))

• \[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{ \sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2 }} \]
• \( -1 \leq r \leq 1 \). Positive \( r \) = positive association; negative \( r \) = negative association.
• \( r = 0 \) means no linear correlation

3. Regression Line (Least Squares)

• Equation: \( y = mx + b \) or \( \hat{y} = a + bx \)
  • Slope: \( b = \frac{ \sum (x_i - \bar{x})(y_i - \bar{y}) }{ \sum (x_i - \bar{x})^2 } \)
  • Intercept: \( a = \bar{y} - b \bar{x} \)

4. Interpreting Regression

• Slope (\( b \)) = change in \( y \) for one unit increase in \( x \)
• Intercept (\( a \)): predicted \( y \) when \( x = 0 \)
• Only valid in the range of observed \( x \) values (avoid extrapolation)

5. Regression & Correlation Analysis

• Assess fit: higher \( |r| \) = stronger linear relationship
• Residual = actual \( y \) - predicted \( y \) (from regression line)
• Standard error, \( r^2 \) (coefficient of determination) measures variance explained

6. Exponential Regression

• Exponential model: \( y = ab^x \), fitted using nonlinear regression methods
• If \( y \) grows/decays multiplicatively, use exponential regression