Fundamental limit to accuracy

Question

In any real experiment, Eventhough we conduct experiment with utmost accuracy some random errors are bound to occur.
$$P\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{\left(x-x_0\right)^2}{2\sigma^2}}$$
This is the Gaussian distribution which defines the probability of getting $x$ in any experiment with mean $x_0$ and standard deviation $\sigma$.
Considering this we get that,
$$P\left(x_0\right)=\frac{1}{\sigma\sqrt{2\pi}}$$
since this is probalility then $P(x_0)\le1$ or,
$$\sigma \ge \frac{1}{\sqrt{2\pi}}$$
What this means is that however nicely an experiment is done the $\sigma$ can never be less than $\frac{1}{\sqrt{2\pi}}$
So is there a fundamental limit to accuracy or I have been wrong in this derivation?
Please Explain?

Answer 1

$P(x)$ is not the probability of getting $x.$ The probability of getting $x$ is zero for any real number. (How would we even know whether we got $x$ or not in a real experiment? It's not like our instruments ever output an infinite number of digits).

$P(x)$ is a probability density function. What it tells you is that for an interval $[a,b],$ the probability that $x$ lies in $[a,b]$ is $$ \int_a^bP(x)dx.$$

As $P(x)$ is not a probability, there is no particular reason it needs to be less than one. In fact, you can see it actually has units of $1/x,$ so it doesn't even make sense to think about its relationship to $1.$

In principle, $\sigma$ can be anything (it also has units of $x$ so it would make no sense for it to be bounded by a dimensionless number). $\sigma$ is just the precision of the measurement. It's possible there's a fundamental limit on this (for physics reasons) but there's no limit set by probability theory.