Common Core Standards: Math
2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
You know the old saying, "A picture is worth a thousand words"? When it comes to math, a picture is probably worth a million. Sometimes (especially when it comes to statistics), one picture or graph can give you a crystal-clear look at what's going on, while a list of numbers just leaves you looking for the nearest exit.
One awesome example of this is a scatter plot. This shows the relationship between two different quantities. It gives you an instant idea of what the data is doing: whether it's grouped in some sort of a pattern or scattered all over the place (hence the name!).
Usually, the data does follow a pattern and fairly often, this pattern takes the shape of a line. Mathematicians, being who they are, immediately want to know which line the data is closest to because it can give us a good approximation of what other data points might be.
The easiest way to find this line is to take a straightedge (a ruler, the edge of folded piece of paper, or our favorite, a piece of uncooked spaghetti) and try to get it to fit through as many of the points as possible. This line is called the line of best fit. (Yeah, every once in a while, mathematicians actually use a name that tells you what the thing is. This is one of those times.)
Students should understand that in many cases, a line of best fit can apply to a scatter plot and provide a good means of understanding the relationship between the variables. They should also be able to guesstimate the accuracy of the data by looking how closely the line of best fit approximates the data points.
Sometimes, the "line" of best fit isn't a line at all, but a curve of some sort. It's possible for the best fit to be a parabola or exponential graph. Just because there isn't a line of best fit doesn't mean the variables aren't somehow related. And if they aren't related for some reason, then no graph—line or curve—will be a "best fit." On the other hand, students should know that enough relationships are linear (or linear enough) to have a line of best fit.