Mean and Standard Deviation Aren’t Just a Summary. They’re an Assumption. | by Matthew Chequers, Ph.D. | Aug, 2026 | Medium
I’m going to start this post with a simple demonstration. Below are density curves for 3 different distributions. The first is a typical bell curve. The middle one is bimodal and has two clusters with a gap — note that the mean sits in the valley where almost no data is present. The last one has a long right tail, with the mean being pulled above the most common value. All 3 datasets have a mean of 10 and a standard deviation of 3. However, if you report “10 ± 3” for any of them, the reader will almost always picture the first plot. Not because they don’t know what they are doing, but because that’s the shape those two numbers typically describe. I have found that most analyses follow a pretty common process: load the data, call describe(), and look at the mean and the standard deviation. They go in the report as x ± y. Everyone nods, and we all move on. However, I think this process glosses over something non-trivial: where do the formulas for mean and standard deviation actually come