Given that measurements are [probabalistic in accordance with the Born Rule](Born%20rule.md) In [quantum mechanics](Quantum%20Mechanics%20(index).md), we may define the _[variance](variance)_ or _uncertainty_ of a [quantum measurement.](Quantum%20measurement%20(index).md) For an [expectation value](expectation%20value.md) $\langle \hat{O} \rangle_{\psi}$ of a [state vector](State%20vector.md), $|\psi\rangle$, we define the variance of an [observable](Observable.md) as $(\sigma_\hat{O})^2 = \langle (\Delta \hat{O})^2 \rangle = \langle \hat{O}^2 \rangle-\langle \hat{O} \rangle^2$ where $\sigma_\hat{O}$ is its [standard deviation](Uncertainty%20(quantum%20mechanics).md) and the observable, $\Delta \hat{O}$, is the [deviation from the mean.](Uncertainty%20(quantum%20mechanics).md#Deviation%20from%20the%20mean) The derivation follows directly from the definition of the [[variance]]. %%do this derivation%% #QuantumMechanics/QuantumMeasurement #QuantumMechanics/QuantumMeasurement/QuantumObservables