Lesson 45: Finding the Line of Best Fit (Exploration: Collecting and Analyzing Data)

New Concept

A line that best fits the points in a scatter plot is a line of best fit.

What’s next

Next, you'll learn to plot data, sketch a line of best fit, and write its equation to make predictions.

Property

A measure of the strength and direction of the relationship between two variables or data sets is called correlation. A relationship can have a positive correlation (points rise), a negative correlation (points fall), or no correlation (points are scattered).

As the hours you study increase, your test scores tend to increase. This is a positive correlation.
The more you play a video game, the less battery life your controller has. This is a negative correlation.
The number of hats someone owns has no relationship to their height. This shows no correlation.

Correlation is like checking if two things are buddies. When one variable changes, does the other change in a predictable way? If they both increase, they're positive pals. If one goes up while the other goes down, they're negative nemeses. If there's no pattern at all, they have no relationship.

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

The strength and direction of a linear correlation is measured by the correlation coefficient, $r$. The values of $r$ can range from $-1$ to $1$. The further $r$ is from $0$, in either direction, the closer the points are to a straight line. When $r$ is zero, there is no linear correlation.

If $r = 0.95$, the data has a strong positive correlation, with points forming a nearly straight line sloping upwards.
If $r = -0.15$, the data has a very weak negative correlation; the points are widely scattered but have a slight downward trend.
An $r$-value of $1$ means all data points lie perfectly on a single line that slopes upwards.

Think of the correlation coefficient, $r$, as a score from $-1$ to $1$ that grades how well your data points form a straight line. A perfect $+1$ or $-1$ means they are perfectly aligned in a positive or negative direction. A score of $0$ means the points are just a random mess with no line.