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.
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.
For what it's worth, you've convinced me that my beloved box plots need to be explained if I want to use them again.
The SVG you've provided clearly shows that the box plot splits the data in 4. The interquartile range (IQR) is clearly marked and it even has a comparison for what the standard deviation (variance) measure would be.
Secondly, if the data truly came from a normal distribution, there are no outliers. Outliers are data points which cannot be explained by the model and need to be removed. Unless you have a good reason to exclude the data points they should be included. This is why I like the IQR and the median, they are not swayed by a few wide valued data points. The 1.5*IQR rejection filter I think is lazy and unjustified. Happy to discuss this point further as it is a bug bear of mine.
Many RTS games have already done this in a traditional client rendered setup with deterministic lockstep (e.g. [1]). The basic idea is to have an input at time t which has a fixed delay until t + x. During [t, t+x] everyone syncs what actions they will take, and so everyone renders the exact same state. Of course, if this sync fails then the game will stutter for everyone, unlike in the stadia case, and it has pros and cons for various types of game. It was largely rejected for action games in the past. The trick is you need to specify the minimum latency that everyone must get under for it to run smooth. In the server-rendering case you would be locked to the renderer's region and would have people with better internet having a strong advantage.
If you're interesting in this sort of thing, check out [2] which goes into depth on how to synchronise game simulations in a bunch of different ways.
It serves to distance it from the moment-based statistics like mean and variance at least.