Lesson 2: Lines of Fit

Property

A line of best fit through a scatter plot can be modeled with a linear equation by selecting two points that the line passes through or nearly passes through.
Use these points $(x_1, y_1)$ and $(x_2, y_2)$ to calculate the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, then substitute the slope and either point into the point-slope form to write the equation of the line of best fit.

Examples

Property

The line of best fit, also known as the regression line, is the specific line that most accurately represents the data in a scatter plot. It is calculated mathematically to minimize the sum of the squared vertical distances from each data point to the line. The equation is typically in the form $y = mx + b$.

Examples

• For a data set relating hours of sunshine ($x$) to plant growth in cm ($y$), the line of best fit might be $y = 0.5x + 2$. This line is the single best summary of the linear trend.
• A scatter plot shows the relationship between a car''s age in years ($x$) and its value in dollars ($y$). The calculated line of best fit could be $y = -2500x + 30000$.
• Unlike a line of fit which can be estimated by eye, the line of best fit is found using a precise formula, often with a calculator or computer software.

Explanation

While a line of fit is a good visual estimate of the trend in a scatter plot, the line of best fit is the most precise version. It is a unique, mathematically determined line that best models the linear relationship between two variables. This line is calculated to be as close as possible to all the data points simultaneously. We often use technology to find the exact equation for the line of best fit, which is also called the regression line.

Property

The correlation coefficient is a value, $r$, between $-1$ and $1$.

• $r > 0$ suggests a positive (increasing) relationship
• $r < 0$ suggests a negative (decreasing) relationship
• The closer the value is to 0, the more scattered the data.
• The closer the value is to 1 or $-1$, the less scattered the data is.

Examples

• The relationship between the distance a car is driven and the amount of fuel used would have a strong positive correlation, with an $r$ value close to 1, like $r = 0.98$.
• Data comparing the number of hours a candle has been burning to its remaining height would show a strong negative correlation, with an $r$ value close to -1, like $r = -0.99$.
• A scatter plot of a person's height versus the last digit of their phone number would have a correlation coefficient very close to 0.

Explanation

The correlation coefficient, $r$, is a score from -1 to 1 that measures how strong a linear relationship is. A score near 1 or -1 means the data points form an almost perfect line, while a score near 0 means no line.