MAT 1000 Mathematics in Today's World

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Today First: a warning about interpreting correlation. We will also talk about least-squares regression. This is a way to calculate the line that is the “best fit” for the data, in other words: a line that is a good approximation of the scatterplot. The reason least-squares regression is important is that it allows us to make predictions where we don’t have any data— these predictions will be based on the pattern the data gives us.

“Correlation is not causation” You may have heard this expression before. What does it mean? Correlation is good evidence for a cause and effect relationship between two variables. If there is such a relationship, the variables will have a strong correlation. On the other hand, variables can have a strong correlation even though there is no cause and effect relationship.

“Correlation is not causation” Example Ice cream sales are correlated with drowning deaths. Obviously not a cause and effect relationship. In this case the explanation is that ice cream sales and drowning deaths are both related to the weather. More ice cream is sold in the summer, and more people go swimming in the summer. We call this relationship between ice cream sales and drowning deaths “mutual response.”

“Correlation is not causation” Correlation may not even be due to mutual response. Example (The Pirate Effect) The number of pirates is correlated with global average temperature: over the past few centuries the number of pirates has decreased, and global average temperatures have increased. Is global warming caused by lack of pirates? This is just a coincidence. People call this kind of relationship a “nonsense correlation.” For more nonsense correlations:

Approximating scatterplots Last time we calculated the correlation between the heights and weights of five male adults. Here is that same data as a scatterplot.

Approximating scatterplots If you had to draw by hand a line that approximated the shape of this scatterplot, you could end up with any number of lines.

Approximating scatterplots For example, maybe you would draw this line

Approximating scatterplots Or this one

Approximating scatterplots But there is only one “least-squares regression line:”

Review of linear functions

Now connect these two points with a line

The least-squares regression line

Note that none of the data actually lies on the line. For a line to be the least-squares regression line the distance from all of the data to the line must be as small as possible. Nevertheless, the line need not (and usually does not) contain any of the data values.

Predictions The most important application of least-squares regression lines is for making predictions. If a scatterplot has a linear form, this suggests an underlying pattern. Mathematically, that pattern is exactly the least-squares regression line. We can then make predictions based on the pattern we see in the data we’ve collected.

Predictions

One danger in using least-squares regression for predictions is extrapolation. Within the range of our data, the least-squares regression line should give reasonable predictions. But, if we plug in numbers too far outside that range, the predictions may no longer be reasonable. In our original height and weight data, the heights range from 67 inches to 77 inches. We can be confident that our least-squares regression line gives reasonable predictions for any height in this range.

Predictions