Why does $\pm$ mean two different things in statistics?

In statistics, why do we sometimes use the symbol $$\pm$$ to mean interval of numbers, for example in confidence intervals, and sometimes to mean two numbers, for example in the formula $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$?

• In both cases the meaning is the same “+ or -“
– Tim
May 7 '21 at 12:38
• Hint: An interval is fully defined by its endpoints, which is two numbers.
– Ben
May 7 '21 at 13:45
• One can say that the confidence interval for an estimate is $(\hat{b}+Z_{\alpha/2} \hat{\sigma}, \hat{b}+Z_{1-\alpha/2} \hat{\sigma})$ xor the confidence limits for an estimate are $\hat{\beta} \pm Z_{1-\alpha/2}\hat{\sigma}$. But the latter is not an interval! That's the only case where I've seen the language abused in probability and statistics. May 7 '21 at 16:28

It doesn't mean two different things. The meaning is the same, $$\pm$$ means "the $$\pm$$ symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, $$+$$ or $$−$$, allowing the formula to represent two values or two equations". "$$a \pm b$$" means "$$a + b$$ or $$a - b$$". Same with $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$, you could replace $$\pm$$ symbol with $$+$$ or $$-$$ to get the two possible answers from the formula.
When defining the interval, this is shorthand for defining its bounds, $$x \pm c$$ means the $$[x - c, x + c]$$ interval bounds. Obviously, this makes sense only for an interval that is symmetric and centered around the origin, it wouldn't make sense for an asymmetric interval (think of skewed distribution).