Lesson 11-2: Draw Lines of Fit

Property

A trend line (or line of best fit/regression line) is a straight line drawn on a scatter plot that models the relationship between two quantitative variables. It provides a one-dimensional summary of a bivariate (2-dimensional) data set by showing the general direction of the data.

To draw it, use the primary method of 'eye-balling' to find the line that minimizes the distance between each data point and that line. A line of best fit will have about the same number of points above and below it and may or may not pass through any of the data points.

Examples

• If the number of hours studied increases and test scores also tend to increase, a trend line would have a positive slope, showing a positive association.
• A scatter plot shows hours spent practicing piano versus number of mistakes made in a performance. The points trend downwards, so an 'eyeballed' line with a negative slope is drawn to show that more practice is associated with fewer mistakes.
• Data is collected on daily temperature and the number of bottles of water sold at a park. The points on the scatter plot go up and to the right. An 'eyeballed' line with a positive slope summarizes this positive association.

Property

When drawing a line of best fit, outliers (points that lie far outside the general pattern) should be ignored. Excluding these points prevents them from unduly determining the slope of the line, ensuring the line accurately models the center of the main data cluster.

Examples

Property

Once a line of best fit is drawn, you can calculate its algebraic equation. First, locate two points that sit exactly on the line of fit to find the slope. Then, use the point-slope form to write the full equation.

Examples

• Given points $(2, 5)$ and $(6, 9)$ on a line of best fit, find its equation.
• First, find the slope: $m = \frac{9-5}{6-2} = \frac{4}{4} = 1$.
• Then, use the point-slope form with point $(2, 5)$: $y - 5 = 1(x - 2)$, which simplifies to $y = x + 3$.

Explanation

Even though a scatter plot is constructed by converting two-variable data into ordered pairs $(x, y)$ and plotting them, the line of fit may or may not pass through any of the actual data points.