Uncertainty in Quantum Mechanics In Quantum Mechanics, as we know, the uncertainty is defined as
$$\Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}.$$
My question is - Why is uncertainty equal to the standard deviation? 
 A: The uncertainty doesn't have to be equal to the standard deviation. Standard deviation however is a frequently used measure of uncertainty in any probabilistic domain (not necessarily QM). All the quantum-mechanical relations involving uncertainties could be derived using another uncertainty measure, although the math involved may be more challenging.
A: For a random variate $x$,
 $$\langle ( x-\langle x\rangle )^2\rangle ^{1/2}$$ 
is by definition its standard deviation*.
A quantum measurement is like any other statistical measurement so the uncertainity being the range in which the observable may lie, $1$ standard deviation is a good statistical measure of uncertainty, though not the only one.

*provided its first and second moments exist.
A: 
In Quantum Mechanics, as we know, the uncertainty is defined as-$$\Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}$$

This is not quantum mechanical uncertainty.

My question is- Why is uncertainty equal to the standard deviation? 

It is not. The quantum mechanical uncertainty in measuring the variable $x$ is given by the probability distribution for measuring the variable  $x$ . The probability  distribution is given by the complex conjugate squared of the wavefunction describing  the measurement of x.
Lets take as an illuminating example the experiment " electrons scattering off a double slit, with given parameters" and ask what is the probability of the single electron to be found at x in this plot:

Take the top frame where single spots mark where an electron has hit. The standard deviation for x, and y are within the micrometer  measurements , the errors on the x. The quantum mechanical uncertainty is seen by the spread of the dots ( footprints of the electrons impinging the same way on the slit). The full complexity of the quantum mechanical uncertainty is seen in the lowest frame, where the wave nature of the solution is also obvious, but also that the quantum mechanical uncertainty is very much different than the statistical measurement uncertainty.  
A: There are many ways to measure the spread of a distribution in statistics apart from standard deviation. For example we also use the interquartile range, frequently regarded as a better measure (depending on the situation) and there are those who advocate the mean absolute deviation.
Standard deviation is often chosen because it has convenient properties; for example, it is a natural parameter for the normal distribution, which, because of the central limit theorem, is often a reasonable approximation to an unknown probability distribution. Whether we use standard deviation, some other measure or something less common, is a matter of human choice.
Likewise, the definition of uncertainty so that it is equal to standard deviation is a matter of human choice. It is a good choice, both because it corresponds to one of the most used statistical measures (i.e. standard deviation), and because it is convenient to work with. Also, by using this particular definition of uncertainty, it is straightforward to prove Heisenberg's uncertainty principle in the usual form.
Again, it is a matter of human choice, but it is certainly a good choice and there is no substantial reason to change it.
