[ColFrancis]

The quartiles are defined by doing the sorting and splitting algorithm. So if you want quartiles (or any other quantile generally) you need to calculate it that way. The mean and standard deviation (sigma) are fundamentally different, which is why the image you linked shows them to contrast against the quantiles.

If you want to represent the standard deviation with your box plot, you can calculate it using standard formulas, many maths libraries have them built in. I don't know how to plot it using any graphing package though. ggplot, plotly and matlab all use the quantiles (the ones I have experience with). Perhaps where ever you learned to read them as mean and standard devation has a reference you could use?

> They are too random to be meaningful. It does not make sense to draw a box plot from that.

This can be a problem. In practice, the distributions I see don't go too crazy and are bounded (production rates can't be negative and can't be infinite). I prefer to use the 10th and 90th percentiles which are well defined and better behaved for most distributions. I do make sure it's very clearly marked on each plot though as it's not standard. Using the 1.5 x IQR cutoff is no better though as when you have enough samples you find that the whiskers just travel out to the cutoff.