This document outlines the least squares method for finding the equation of a linear regression line that best fits a set of data points. It provides the mathematical formulas for calculating the slope (a) and y-intercept (b) of the line. As an example, it analyzes turnover data from 2007 to 2011 for a company to estimate turnover in 2012. The slope is calculated as 79.4 and the y-intercept as 1.8, providing the final estimated linear regression equation of y = 79.4x + 1.8 to predict a 2012 turnover of around 478,200 euros.

1. THE LEAST SQUARE METHOD
Introduction:
First of all, you must find the figures of your company: the turnover (CA) of the
last 5 years. Then add them in a data table.
For example:
Years 2007 2008 2009 2010 2011 1
Turnover K€ 80 162 250 300 408
Draw an adjusted line on a chart with:
- The years in abscissa (xi)
- The turnover in ordinate (yi)
Adjusted line
450
400
350
300
250
200
150
100
50
0
2006 2006,5 2007 2007,5 2008 2008,5 2009 2009,5 2010 2010,5 2011
The purpose of this method is to find statistically the equation of the line
y = ax + b since a data table.

2. I. The mathematical formulas
 a = ∑ xi yi – n x y
∑ xi² - n x ²
a equals sum of xi times yi minus n times the x values times the y values
divided by sum of xi exponent 2 minus n times the x values exponent 2
2
 b=y–ax
b equals the y values minus a times the x values
n means the number of years taken in account
II. The data table and calculations
xi yi xi² xiyi
1 80 1 80
2 162 4 324
3 250 9 750
4 300 16 1200
5 408 25 2040
∑ 15 ∑ 1200 ∑ 55 ∑ 4394
 In column 1 of the table: xi
You transform the years in numbers. For example:
2007 = 1
2008 = 2
2009 = 3 etc…

3.  In column 2 of the table: yi
You enter the turnover in K€
 In column 3 of the table: x²
You multiply: xi xi
 In column 4 of the table, you multiply : xi yi
At the ends of each column, calculate its sum. Then, 3
 You calculate:
x = ∑ xi / n = 15/5= 3
the x values equal the average of xi
y = ∑ yi / n = 1200/5= 240
the y values equal the average of yi
III. Solving of the equation y = ax + b with the mathematical formulas
a = 4394 – ( 5 x 3 x 240)
55 – (5 x 3²)
a = 794 = 79,4
10
b = 240 – (79,4 x 3)
b = 1,8

4. The equation of the adjusted line is:
y = ax +b
y = (79,4 x 6) + 1,8
IF
2007 = 1 4
2008 = 2
2009 = 3
2010 = 4
2011 = 5
THEN 2012 = 6
y = 478,20
y = the estimated turnover for 2012 is 478 200 €