Least-Squares Line

Definition

A least-squares line, also known as the line of best fit, is a straight line that minimizes the sum of the squared differences between observed values and the values predicted by the line. It is used in linear regression to model the relationship between two variables.

5 Must Know Facts For Your Next Test

1. The equation of a least-squares line is typically written as $\hat{y} = b_0 + b_1x$, where $\hat{y}$ represents predicted values, $b_0$ is the y-intercept, and $b_1$ is the slope.
2. The least-squares method minimizes the sum of squared residuals, which are the vertical distances from each data point to the line.
3. The slope ($b_1$) indicates how much $\hat{y}$ changes for a one-unit change in $x$.
4. The intercept ($b_0$) represents the predicted value of $\hat{y}$ when $x = 0$.
5. Calculating a least-squares line involves using formulas: $b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ and $b_0 = \bar{y} - b_1\bar{x}$.

Review Questions

• What does the slope ($b_1$) of a least-squares line represent?
• How do you interpret the y-intercept ($b_0$) in a regression equation?
• What mathematical method does a least-squares line use to fit data points?

Related terms

Residual: The difference between an observed value and its corresponding predicted value on a regression line.

Correlation Coefficient: $r$, a statistical measure that describes the strength and direction of a linear relationship between two variables.

Coefficient of Determination: $R^2$, a measure that shows how well data points fit a statistical model – specifically, it indicates what proportion of variance in dependent variable is predictable from independent variable.