Linear growth means that the relationship between the variables is a straight line relationship. An example of a linear relationship is the relationship between temperature in Fahrenheit and Celsius is F=(9/5)*C + 32. Here the slope (or the rate at which F changed when C changes one unit) is 9/5 and the intercept (or displacement in F when C is zero) is 32. Here we have expressed the model for this relationship as a formula. A graphical expression of this relationship is given on the right. A table representing the function is given below.
The highlighted points on the above graph are the points from the table.
If we did not already know this relationship, we would collect data using two thermometers (one Fahrenheit and one Celsius), make a table like the one on the left, and then plot the points and connect them with straight line segments. From the graph we could then estimate the intercept and the slope of this function and thus write the formula. In fact that is the way the new thermometers or thermocouples are calibrated.
| C | F |
| -10 | 14 |
| 0 | 32 |
| 10 | 50 |
| 20 | 68 |
| 30 | 86 |
| 40 | 104 |
| 50 | 122 |
| 60 | 140 |
| 70 | 158 |
| 80 | 176 |
| 90 | 194 |
| 100 | 212 |
| 110 | 230 |
Lets explore this function a little more.
First from the formula standpoint, the derivative of F with
respect to the variable C is given by 
which is
the slope of our line.
Secondly, suppose we add another column to our table by
computing the difference between successive values of F and
dividing by the difference between the related values of C. Thus
the first element in the new column would be 
which is an
approximation to the slope of the line. Notice that this value is
constant throughout the table (we have a straight line, constant
slope).
| C | F |  F/ C
|
| -10 | 14 | |
| 0 | 32 | 1.8 |
| 10 | 50 | 1.8 |
| 20 | 68 | 1.8 |
| 30 | 86 | 1.8 |
| 40 | 104 | 1.8 |
| 50 | 122 | 1.8 |
| 60 | 140 | 1.8 |
| 70 | 158 | 1.8 |
| 80 | 176 | 1.8 |
| 90 | 194 | 1.8 |
| 100 | 212 | 1.8 |
| 110 | 230 | 1.8 |
How do you fit the equation of a line to a set of real data that contains variation or error in the measured values? In statistics this is called simple linear regression. What we want to do is find values for the intercept and the slope that minimize the sum of the squared deviations of the y's about the line.