On the whole it seems like a nice idea, but there's a couple of weird things, such as:

> Note: If you're curious why there is a negative number (-5) in the histogram, that's just an inevitable downside of the simplicity of the Unsure Calculator. Without further knowledge, the calculator cannot know that a negative number is impossible (in other words, you can't have -5 civilizations, for example).

The input to this was "1.5~3 x 0.9~1.0 x 0.1~0.4 x 0.1~1.0 x 0.1~1.0 x 0.1~0.2 x 304~10000" - every single range was positive, so regardless of what this represents, it should be impossible to get a negative result.

I guess this is a consequence of "I am not sure about the exact number here, but I am 95% sure it's somewhere in this range" so it's actually considering values outside of the specified range. In this case, 10% either side of all the ranges is positive except the large "304~10000".

Trying with a simpler example: "1~2 x 1~2" produces "1.3~3.4" as a result, even though "1~4" seems more intuitive. I assume this is because the confidence of 1 or 4 is now only 90% if 1~2 was at 95%, but it still feels off.

I wonder if the 95% thing actually makes sense, but I'm not especially good at stats, certainly not enough to be sure how viable this kind of calculator is with a tighter range. But just personally, I'd expect "1~2" to mean "I'm obviously not 100% sure, or else I wouldn't be using this calculator, but for this experiment assume that the range is definitely within 1~2, I just don't know where exactly".

>The input to this was "1.5~3 x 0.9~1.0 x 0.1~0.4 x 0.1~1.0 x 0.1~1.0 x 0.1~0.2 x 304~10000" - every single range was positive, so regardless of what this represents, it should be impossible to get a negative result.

Every single range here includes positive and negative numbers. To get the correct resulting distribution you have to take into account the entire input distribution. All normal distributions have a non-zero possibility to be negative.

If you want to consider only the numbers inside the range you can look at interval arithmetic, but that does not give you a resulting distribution.

The calculator in Emacs has support for what it is you request, which it calls "interval forms". Interval form arithmetic simply means executing the operations in parallel on both ends of the interval.

It also has support for "error forms" which is close to what the calculator in OP uses. That takes a little more sophistication than just performing operations on the lower and upper number in parallel. In particular, the given points don't represent actual endpoints on a distribution, but rather low and high probability events. Things more or less likely than those can happen, it's just rare.

> I'm not especially good at stats

It shows! All the things you complain about make perfect sense given a little more background knowledge.

> every single range was positive, so regardless of what this represents, it should be impossible to get a negative result.

They explain that the range you give as input is seen as only being 95% correct, so the calculator adds low-probability values outside of the ranges you specified.

I can see how that surprises you, but it's also a defensible design choice.