If we quote $a = 1 \pm 0.5$, we usually mean that the probability for the *true* value of $a$ is Gaussian $G(a, \mu, \sigma)$ distributed with $\mu = 1$ and $\sigma = 0.5$.
"let $X_{1},X_{2},\dots ,X_{n}$ denote a statistical sample of size $n$ from a population with expected value (average) $\mu$ and finite positive variance $\sigma ^{2}$, and let $\bar {X_{n}}$ denote the sample mean (which is itself a random variable). Then the limit as $n\to \infty$ of the distribution of $\frac {({\bar {X}}_{n}-\mu )}{\frac {\sigma }{\sqrt {n}}}$, is a normal distribution with mean 0 and variance 1."