How To Calculate Regression Coefficient

One the most basic tools for engineering or scientific analysis is linear regression. This technique starts with a data set in two variables. The independent variable is usually called "x" and the dependent variable is usually called "y." The goal of the technique is to identify the line, y = mx + b, that approximates the data set. This trend line can show, graphically and numerically, relationships between the dependent and independent variables. From this regression analysis, a value for correlation is also calculated.

Step 1

Identify and separate the x and y values of your data points. If you are using a spreadsheet, enter them into adjacent columns. There should be the same number of x and y values. If not, the calculation will be inaccurate, or the spreadsheet function will return an error. x = (6, 5, 11, 7, 5, 4, 4) y = (2, 3, 9, 1, 8, 7, 5)

Step 2

Calculate the average value for the x values and the y values by dividing the sum of all the values by the total number of values in the set. These averages will be referred to as "xavg" and yavg." xavg = (6 + 5 + 11 + 7 + 5 + 4 + 4) / 7 = 6 yavg = (2 + 3 + 9 + 1 + 8 + 7 + 5) / 7 = 5

Step 3

Create two new data sets by subtracting the xavg value from each x value and the yavg value from each y value. x1 = (6 – 6, 5 – 6, 11 – 6, 7 – 6 ... ) x1 = (0, -1, 5, 1, -1, -2, -2) y1 = (2 – 5, 3 – 5, 9 – 5, 1 – 5, ... ) y1 = (-3, -2, 4, -4, 3, 2, 0)

Step 4

Multiply each x1 value by each y1 value, in order. x1y1 = (0 -3, -1 -2, 5 * 4, ... ) x1y1 = (0, 2, 20, -4, -3, -4, 0)

Step 5

Square each x1 value. x1^2 = (0^2, 1^2, -5^2, ... ) x1^2 = (0, 1, 25, 1, 1, 4, 4)

Step 6

Calculate the sums of the x1y1 values and x1^2 values. sumx1y1 = 0 + 2 + 20 – 4 – 3 – 4 + 0 = 11 sumx1^2 = 0 + 1+ 25 + 1 + 1 + 4 + 4 = 36

Step 7

Divide "sumx1y1" by "sumx1^2" to get the regression coefficient. sumx1y1 / sumx1^2 = 11 / 36 = 0.306