Line of Best Fit(Least Square Method)

A line of best fit is a straight line that is the best approximation of the given set of data.

It is used to study the nature of the relation between two variables.

A line of best fit can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).

A more accurate way of finding the line of best fit is the least square method .

Use the following steps to find the equation of line of best fit for a set of ordered pairs.

Step 1: Calculate the mean of the x-values and the mean of the y-values.

Step 2: Compute the sum of the squares of the x-values.

Step 3: Compute the sum of each x-value multiplied by its corresponding y-value.

Step 4: Calculate the slope of the line using the formula:

  

where n is the total number of data points.

Step 5: Compute the y-intercept of the line by using the formula:

where are the mean of the x- and y-coordinates of the data points respectively.

Step 6: Use the slope and the y -intercept to form the equation of the line.

Example:

Use the least square method to determine the equation of line of best fit for the data. Then plot the line.

Solution:

Plot the points on a coordinate plane.

Calculate the means of the x-values and the y-values, the sum of squares of the x-values, and the sum of each x-value multiplied by its corresponding y-value.

Calculate the slope.

Calculate the y-intercept.

First, calculate the mean of the x-values and that of the y-values.

Use the formula to compute the y-intercept.

Use the slope and y-intercept to form the equation of the line of best fit.

The slope of the line is –1.1 and the y -intercept is 14.0.

Therefore, the equation is y = –1.1 x + 14.0.

Draw the line on the scatter plot.