Here are 12 + 4 famous datasets plotted (click to get an enlarged version).
The main idea is that the correlation coefficient is equal within the sets.

This means that a fit of a model to the data without evaluating the printout, investigating residuals or graphs will miss important information. This type of analysis might be a problem if programmatically carried out by mass-analysis.

Dataset 1-13        Dataset 14-17

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A simple regression model

Correlation coefficient – a simple but sometimes misused statistic

An 'ordinary-least-squares (OLS)' model

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More regression (Sw)