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 _[standard deviation](standard%20deviation)_ 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 uncertainty of an [observable](Observable.md) as $\sigma_\hat{O} = \sqrt{\langle (\Delta \hat{O})^2 \rangle} =\sqrt{\langle \hat{O}^2 \rangle-\langle \hat{O} \rangle^2}$ where $\Delta \hat{O}$ is an observable that gives the [_deviation from the mean_.](Uncertainty%20(quantum%20mechanics).md#Deviation%20from%20the%20mean) # Deviation from the mean The _deviation from the mean,_ $\Delta \hat{O}$ is a [Hermitian operator](Hermitian%20operators.md) dependent on the [[State vector]] that represents the deviation from the mean. Thus it is written as $\Delta \hat{O} = \hat{O} - \langle \hat{O} \rangle_{\psi}.$ ^3cb055 #QuantumMechanics/QuantumMeasurement #QuantumMechanics/QuantumMeasurement/QuantumObservables