Linear Regression is a process by which the
equation of a
line is found that “best fits” a given set of data. The
line of best fit approximates the best linear representation for your data. One very important aspect of a regression
line is the relationship between the
equation and the “science quantity” often represented by the
slope of the line.
Examples of how linear regression is used in a science application can be seen in
PhysicsLAB. Worksheets that accompany this lesson can be located under related documents, worksheets,
Data Analysis #1-#8.
Let's look at an example of linear regression by examining the
data in the following
table to discover the relationship between temperatures measured in Celsius (Centigrade) and Fahrenheit. [Remember that lines are named using the convention
y vs. x whereas
data tables are constructed as
x | y.]
Based on this data:
- interpolate the equivalent temperature in degrees Celsius of our body temperature, 98.6 °F
- relate the linear equation of your model with its associated science formula to determine the "physical" meaning of the slope of this data's trend line
Step 1: First we will plot the
data using a TI-83 graphing calculator. We will
enter the data measured in degrees Fahrenheit in L1 and the temperatures measured in degrees Celsius in L2. Once the
data is entered, your screen should look like the following:
After entering the
data into the calculator,
graph the data. The Fahrenheit data, listed in L1, represents the x-axis, and the Celsius data, listed in L2, represents the y-axis. Your screen should look like the following:
Step Two: Now we need to find a
linear equation that models the
data we have plotted. According to the calculator, our
equation has the following properties:
Based on the
graph and the
equation information listed above, our correlation
coefficient (r) is equal to 1. That means that our
data perfectly models a linear function.
Step Four: Using the model from step two and the
graph on our calculator from step three, we can trace along the
graph and determine what temperature in degrees Celsius equals 98.6 °F, our body temperature.
This screen capture shows us that 98.6 °F (x-value) is equivalent to 37 °C (y-value).
Step Five: Consider our
equation 
. The accepted formula used to convert Fahrenheit degrees to Celsius degrees is typically written as
Expressed in this form we can clearly see that our model's
equation is indeed the same
equation conventionally used to convert temperatures between these two measuring scales. Since our line's
slope [
] is a decimal, we know that the size of a "degree" on the two temperature
scales is not the same; that is, these
scales are not in a one-to-one correspondence -- 1 Celsius degree (Cº) does not equal 1 Fahrenheit degree (Fº).
Examining the
slope in
fraction form [
] we can clearly see that the relationship between the two
scales is such that from a given
point on the line, you move up five degrees on the Celsius
scale and right nine degrees on the Fahrenheit
scale to arrive at the next
point on the line. Or equivalently, when the temperature changes 9 Fº it only changes 5 Cº.