Least Squares

1. By: Sean DuBois Least Squares

2. What are Least Squares? • The method of least squares attempts to answer what linear line the data lies on. It can also be used to predict values. • In other words it attempts to find the line of best fit.

3. Uses • Can be used to predict: • gas prices • health care expenditures • rate of inflation • number of cars used in a given country • Disease death rates • Percentage of people completing higher education • Etc

4. Squares Error To measure the error of the line, the sum of all points on the line is shown by the equation: + . . . .+ E is used to observe points that are fitted to the line. If the point lies on y=mx+b then the error of Ei is zero. If the given point is farther away from the line then the error of E will be higher.

5. Continued • In most cases the error will never be zero. The error can only be zero if the data points lie directly on a straight line, so in that case you need to figure the least squares line with the following equation:

6. Equation Continued • ∑x= sum of the x coordinates of the data points. • ∑y= sum of all the y coordinates • ∑xy= sum of the products • ∑ = sum of the squares of the x coordinates • N= number of data points

7. Example • The following data will be examined using least squares:

8. Example Continued • The first step involves finding the slope and y intercept of the data points supplied on the previous slide. • Once these have been found you can graph the data points and find the line of best fit.

9. Example Continued

10. Analysis • Looking at the graph you can see that gas prices are declining in New York City. The trend line shows this as it is linearly decreasing because of a negative slope.

11. Conclusions • Least Squares can be a simple method of figuring out trend lines by using data points. • It can be used to predict future prices of say gas or stocks. • With the example it showed that gas prices do not follow a linear trend because of outside variables such as a healthy economy.