Scatter Plots and Models
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Before attempting a regression analysis of data, it is often helpful to examine a scatter plot of the data to see which regression model is most likely going to be a good fit.  Keep in mind that when working with real world data, it is unlikely that any regression model is going to be a "perfect" fit.  The goal is to find the model that fits as many of the data points as possible and will be the best indicator of trends in the data. 
 

A scatter plot graphically displays two related sets of data.  Such a visual representation can indicate patterns, trends and relationships.

On this page, we will be concentrating on examining scatter plots to determine whether a linear, logarithmic, exponential or power regression model would be most appropriate.

Using the Calculator
 with Scatter Plots

See how to use your TI-83+/TI-84+ graphing calculator  with scatter plots.
Click calculator.

Basic things to look for when looking for a model:
If the "shape" of more than one model appears to fit the data, test all of your choices to see which model is actually the best fit and the best predictor. 
Remember that other representations of these shapes may also exist due to the different natures of the data.
(The following chart is used with permission from MathBits.com.)

linear
y = a + bx
logarithmic
y = a + blnx
exponential
y = abx
power
y = axb

Does the plotted data resemble a straight line?

The slope may be either positive or negative.
Linear associations are the most popular because they are easy to read and interpret.


Does the plotted data ascend rapidly at the left but level off toward the right?

Remember the shape of the natural logarithmic function crossing the x-axis at one and domain x > 0.

Does the plotted data appear to grow (or decline) by percentage increases (decreases)?

Often deals with growth of populations, bacteria, radio-active decay, etc.

Remember the shape of the exponential function, crossing the y-axis at one and range y > 0


Does the plotted data possess characteristics not seen in the first three models?  Not a straight line, but a more gradual change than exponential?

  Power functions are of the form y = axbRemember the nature of such graphs when the exponent is odd and even.
First quadrant:
Outside first quadrant:



Examples: 
Which type of function (linear, exponential, logarithmic, or power) would best model the data in each of the scatter plots shown below?  Explain your reasoning.

1. Linear:
This scatter plot appears to be linear.  The plots could be described as being clustered about a straight line with a negative slope.  There is no obvious "curving" to the plots.
2. Power:
This scatter plot appears to be a power function.  While having "some" tendency toward a straight line, the plot appears to be a closer match to a graph such as which would make it a power function.
3. Exponential:
This scatter plot appears to be an exponential function.  The graph is changing by percentages of decrease.  The domain includes both positive and negative values, the range is greater than zero, and the y-intercept is approximately one.  This plot appears to resemble an exponential rate of decay graph of the form .