Approximate Linear Relationships

In this section, we deal with relationships that do not follow an exact linear relationship. There are many instances in nature, business, and countless scenarios where we find a relationship that is almost linear, but has variation. In order to create an equation that approximately describes a situation, we can start by creating a graph of data points. This is called a scatter plot. We sill see that even the best linear models will have some variation. This deviation is due to many factors, many of which we can not control, but may also be due to factor we can change such as measurement error. A plot that doesn’t seem to fit the data, when most others do is called an outlier.

To make a scatter plot, we create a graph with given points:

Make a scatter plot of the following ordered pairs:
Plot each point individually:

We can see that the points in the previous example are roughly in a straight line. This makes it a reasonable approximate linear relationship. In fact, we can draw a line through the points to show this approximate relationship. We call this line the “best fit” line, or the line of “regression”.

Make a scatter plot of the following ordered pairs and add a best fit line
Begin by plotting each point: To add the trendline, we place it through the middle of the points, in an effort to minimize the total distance of the points from the line.

Two scientist (Margarette Brooks and Amos Dolbear) counted the number of cricket chirps heard on various days, recorded the results, and also recorded the temperature that day. A table of such findings is posted below. Create a scatter plot with “chirps counted in 15 seconds” as the axis and temperature as the axis. Decide if the data is linearly related.

The graph of this data is as follows: The graph looks very linear. In fact, applying the best fit line, we have: We will not go into the details of finding the exact line of best fit in this course. This will come later in a statistics course for most students. For now, it will suffice to have the lines of best fit in the correct general direction, and going through what appears to be the “middle” of the points.

The two scientists of the previous example found that there is a relation between the frequency of cricket chirps and the current temperature. They even created the equation of the best fit line. It is often given as , where T is the temperature, and is the number of chirps counted in 15 seconds.

We can use the best fit line to predict values. In the cricket example, we saw that the linear regression line could have the equation , thus, if we counted 25 chirps in 15 seconds, we could input 25 into the equation to estimate the temperature.

Using the equation , estimate the temperature for 25 cricket chirps counted in 15 seconds.
We simply evaluate     Checking this value on the scatter plot, we can see that the values would fit very nicely with the data.

Estimating a value this way is called interpolation as long as the value being estimated falls within the experimental data (the data that has been collected). We can reasonably assume that our estimations are useful as long as this is the case.

When the value we wish to estimate falls outside the experimental data, we call this extrapolation. One must be extremely careful when extrapolating data. When testing data outside the known values, it is quite often the case that the trendline no longer fits outside the range of collected values.

An outlier is a data point that seems to not fit the data, and skews the line of best fit.

A teacher tests her students reading level (on a scale from 1-20), and their IQ score and finds the data listed below. Make a scatter plot to see if the data is linearly related.

plotting this data, we have: we can see that this data is approximately linear. Adding the trendline, we see this more clearly: But it happens that a student was absent the day they were tested. This particular student, who is often absent, was tested, and added to the data. His IQ was tested at 122, and his reading score was tested at 2. Adding this point to the data we have the plot: When the new trendline is added to the new data: It is apparent that the new point has significantly altered the best fit line. It has pulled it so much, that the line is no longer a good predictor of the relationship. When this happens, we tend to eliminate the outlier from the data, and consider it to either be a fluke, or to have special factors or circumstances that effect it. – In this case, the student in question was autistic. He was particularly brilliant in mathematics and science, enabling him to score quite well in the IQ test, but could not read past a second grade level. Since he is in a fairly different situation than the rest of the class, it seems unreasonable to include his data with the rest of the class. We would want to consider his scores with other students like him in an entirely new set of data.