Least-Squares Regression

Regression is a technique used to predict future values based on known values. For instance, linear regression allows us to predict what an unknown Y value will be, given a series of known X and Y’s, and a given X value.

Given the following, it’s easy to see the pattern. But assuming no obvious pattern exists, regression can help us determine what the Y value will be given our known X values.

X	Y
2	3
4	6
6	9
8	12
10	15
12
14

 

The X value is known as the independent variable, the “predictor variable”, while the Y value is the value you’re being predicted.

The linear regression (or “least squares regression”) equation is Y’ = a + bX

• Y’ (Y-prime) is the predicted Y value for the X value
• a is the estimated value of Y when X is 0
• b is the slope (the average change in Y’ for each change in X)
• X is any value of the independent variable

There are additional formulas for both a and b.

Let’s take a look at the following data-set, that compares the number of calls made for a product against the number of sales:

Calls (X)	Sales (Y)
20	30
40	60
20	40
30	60
10	30
10	40
20	40
20	50
20	30
30	70
220	450

 

First we need to calculate the sum of X-squared, Y-squared and X*Y:

	Calls (X)	Sales (Y)	X2	Y2	XY
	20	30	400	900	600
	40	60	1600	3600	2400
	20	40	400	1600	800
	30	60	900	3600	1800
	10	30	100	900	300
	10	40	100	1600	400
	20	40	400	1600	800
	20	50	400	2500	1000
	20	30	400	900	600
	30	70	900	4900	2100
Total	220	450	5600	22100	10800

 

Returning to our formula, let’s start with b first:

The top of the equation looks like this: b = 10(10800) – 220 * 450 / n(∑X²)-(∑X)². We’ve simply filled in the values from our chart.

b = 10(10800) – 220 * 450
b = 108,000 – 99,000
b = 9,000 / n(∑X²)-( ∑X)²

Now we have to do the bottom half of the equation:

n(∑X²)-(∑X)²

=10(5600)-(220) ²
=56,000 – 48,400
=7,600

Returning to our equation:

b = 9,000 / 7,600
b = 1.1842

Now let’s move on to a:

a = 450 / 10 – 1.1842 * (220 / 10)
a = 45 – (1.1842 * 22)
a = 45 – 26.0524
a = 18.9476

So, going back to our original regression equation, Y’ = a + bX and plugging our numbers, we get:

Y’ = 18.9476 + (1.1842)X

To use this equation, we now put our desired value in for X. With an estimated 20 calls:

Y’ = 18.9476 + (1.1842)*20
Y’ = 18.9476 + 23.684
Y’ = 42.63

So, a salesperson who makes 20 calls will expect to make 42 sales.