A curve of best fit (the word curve includes lines) is the curve that best fits a set of coordinate plane points.

Line of Best Fit

We are often asked to find a line of best fit for a set of plotted points.  That is, we want to find a straight line that appears to flow through the points; we want an equal number of points above and below the lines.  This can be useful in statistics when we wish to determine whether two sets of points show correlation.

To find a line of fit, we plot the data points to form a scatter plot.  We then look at the data to see if it seems to be rising or falling (it is sometimes helpful to use a thin straight edge of some kind; an uncooked piece of spaghetti is helpful!).  We can then manually sketch a line.  We can find two points on the line we draw and then use the point-slope formula to find the equation of the line.

The above process finds a line of fit, but does not necessarily find the line of best fit; different people may find different line estimates.  Statistically, we can find the line that best fits the data through a process called linear regression.  The process and formula for linear regression is out of scope for this wiki, but we can use calculators with statistics capability to perform linear regression and give us the line of best fit.

This "cheat sheet" provides the calculator steps necessary if using a TI-84 to perform linear regression to give us the line of best fit between two sets of data.  Note that the instructions will produce a graph containing the points on a scatter plot with the line of best fit drawn through them.  You may take a short cut by following steps 1, 3, and 4 (press enter at this point).  Basically, you enter the x data in list L₁ and the y data in list L₂ (or any other list of your choice).  We then choose STAT -> CALC -> 4:LinReg(ax+b).  We enter the list names and press enter.  When enter is pressed, you will get the value of the slope (a) and y-intercept (b) of the line of best fit so that you can write its equation in slope-intercept form.

These class notes provide further explanation and examples of finding line of best fit.

Curve of Best Fit

A curve of best fit, like a line of best fit, tries to fit the best curve to a set of data points plotted on a scatter plot.  For example, we may look at a set of plotted points and visualize a parabola that might fit through it.  We can use a quadratic regression in this case.  The mathematical process of performing quadratic regression is out of scope of this wiki, but we can again use calculators with statistics capability.  If using a TI-84 calculator, the process is similar to finding a line of best fit.  We enter the x data in list L₁ and the y data in list L₂ (or any other list of your choice).  We then choose STAT -> CALC -> 5: QUADREG.  We enter the list names and press enter.  This will give us the values of a, b, and c for the second-degree equation ax² + bx + c.

Statistics defines other types of regressions that can be performed if the curve of best fit is another kind of curve, such as a cubic (third degree polynomial) or ellipse.  These kinds of regressions are out of scope of this wiki, but feel free to explore them on your calculator.