[pocketsand]

As I'm sure you know, there are a lot of variations on how quantiles are calculated in various software. The 25th percentile, e.g., doesn't always line up with a value in the dataset, so sometimes nearest rank methods are used, otherwise a linearly interpolated data point, where interpolation is done in various ways.

In any event, none of these methods assume normality, or rely on CDFs of a normal curve.

If they did, every box plot would be symmetric.

The fact some people think that boxplots are constructed in such a way is a pretty good reason to take the author's article seriously as for how boxplots are confusing.

[ColFrancis]

As a first pass definition it does well to explain the concept. Even if you're interpolating you will need to rank the samples and find the two nearest neighbours to interpolate between.

It serves to distance it from the moment-based statistics like mean and variance at least.