Correlation

The correlation coefficient is a measure of the strength and direction of the linear relationship between two quantitative variables. It is denoted as r, and is always between -1 and 1. Here are some examples of scatterplots and their corresponding correlation coefficients.

A visualization of correlations.

As you can see, in the first row of the examples above, the closer the points are to lying on a straight line, the closer the correlation is to either 1 or -1. If the scatter plot has a positive direction, the correlation is a positive number, and if the scatter plot has a negative direction, the correlation is a negative number.

Correlation only measures the strength of linear relationships between variables. The last row of examples shows a variety of scatter plots where there is clearly an interesting relationship between the two variables (note all the unique shapes!), but the correlation is 0 because the relationship is nonlinear. Read this for more detail about how correlation is calculated.

Multiple scatter plots for the questions below.

Question: Using the above figure as a guide, match the correlation to each of the scatterplots from previous questions.

You can use Sheets to find correlation using the CORREL function.

Video - how to find correlation in sheets.

A common, related value is r^2, called the coefficient of determination. **The coefficient of determination is the proportion of variation explained by the explanatory variable. **It can be calculated by squaring the correlation coefficient. The closer r^2 is to 1, the closer r was to either 1 or -1, and thus the stronger the relationship between the variables. The coefficient of determination is useful when you’re only interested in strength, rather than strength and direction.

../_images/scatter-correlation-graph-1.png ../_images/scatter-correlation-graph-2.png ../_images/scatter-correlation-graph-3.png

Q-2: Which would have the largest \(r^2\) value?

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