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.