You will use a spreadsheet to create linear models for data that are not exactly linear, and then interpret the slope of the trendline.
After completing this tutorial, you will be able to complete the following:
When observing real-world data, we may generalize trends we see in the data. We also observe how the values of a dependent variable rely on the values of an independent variable. The dependent variable typically is the y variable, while the independent variable usually is the x variable. Several different mathematical models may accurately describe the relationship between these variables, but sometimes the data are related by a linear relationship. That is, there appears to be a constant rate of change between the two quantities.
However, when we create a scatter plot of real-world data, the data points do not always lie perfectly on a straight line. This is because all measurements are subject to random error. We often are limited in precision due to human error, so the values we observe are not always exact.
For example, consider the following data describing the weight of a car (in pounds) and the miles per gallon that the car gets.
From this data set, we can create a scatter plot.
The graph seems to indicate that as the weight of a car increases, the car's miles per gallons decrease. Although these data do not lie exactly on a straight line, we can still draw a trendline that best fits the data points. The trendline that these data appear to follow is a line with a negative slope.
To find the actual equation of this line in the form , we must rely on extensive calculations, though. Alternatively, we can enter the given data into a spreadsheet that can construct a linear data model.
The reason for the difference in these numbers, called a residual, is that the algorithm used to construct the linear data model relies on minimizing the distance between every data point and the line. This algorithm results in a best fit line,, or a regression line.
Note that in this particular example, the linear data model implies that a car that weighs 0 pounds gets 51.1706 miles per gallon. The estimated value of the dependent variable seems illogical. But asking how many miles per gallon a car that weighs 0 pounds gets is not a reasonable question either. When interpreting and using a linear data model, it is important to only focus on the scope of the model. Beyond the observed values, the real-world data values may drastically change the model that is calculated, resulting in a completely different line or even something more complex, such as a polynomial function.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should be able to compute and interpret the slope of a linear function as a rate of change, understand that a linear function is a function with a constant rate of change, and translate between the graph and equation of any form for linear models.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||linear data model, linear equation, mathematical model|