Random Thoughts in a Random Universe

How Much Do My Measurements Disagree?

Or why “The error bars overlap” is a meaningless statement.

In my work as an astrophysicist I have encountered quite a bit of confusion on how to quantify the level of disagreement between two or more measurements with error bars, or even more complex multi-dimensional confidence volumes. This is even more true this term when I am teaching an undergraduate lab course. Thinking a little about this myself, I realized that this is not something I have ever seen written up. At least not in any concise form. So I decided to create this little document, not only to educate others, but also to organize my own thoughts. I put an emphasis on developing the concepts and I make no attempt at rigour.

Suppose we have three measurements of the gravitational acceleration on the Earth’s surface, all with their own error bars (or measurement uncertainties, if you prefer). These are shown in the plot below and the error bars indicate their $1\sigma$ standard deviation.