Variance and Standard Deviation

Formula

For a state $\rho$ and self-adjoint observable $A$, define the centered observable

$$\delta_\rho A \equiv A-\langle A\rangle_\rho I.$$

The variance and standard deviation are

$$\operatorname{Var}_\rho(A) \equiv (\Delta_\rho A)^2 = \left\langle (\delta_\rho A)^2 \right\rangle_\rho,$$ $$\Delta_\rho A = \sqrt{\operatorname{Var}_\rho(A)}.$$

When the second moment exists,

$$\operatorname{Var}_\rho(A) = \langle A^2\rangle_\rho - \langle A\rangle_\rho^2.$$

Variance has the squared units of $A$; standard deviation has the same units as $A$.

At a glance

Setting	Variance
Centered observable	$\operatorname{Var}_\rho(A)=\langle(A-\langle A\rangle_\rho I)^2\rangle_\rho$
Raw moments	$\operatorname{Var}_\rho(A)=\langle A^2\rangle_\rho-\langle A\rangle_\rho^2$
Pure-state norm	$(\Delta_\psi A)^2=\lVert(A-\langle A\rangle_\psi I)\lvert\psi\rangle\rVert^2$
Density operator	$(\Delta_\rho A)^2=\operatorname{Tr}(\rho A^2)-[\operatorname{Tr}(\rho A)]^2$
Discrete outcomes	$(\Delta A)^2=\sum_a(a-\langle A\rangle)^2p(a)$
Continuous outcomes	$(\Delta A)^2=\int(\lambda-\langle A\rangle)^2p_A(\lambda)\,d\lambda$
Affine transformation	$\operatorname{Var}(\alpha A+\beta I)=\lvert\alpha\rvert^2\operatorname{Var}(A)$ for real-valued observables
Two outcomes $a,b$	$\operatorname{Var}(A)=p(1-p)(a-b)^2$
Bounded spectrum $[a_{\min},a_{\max}]$	$0\leq\operatorname{Var}(A)\leq(a_{\max}-a_{\min})^2/4$

The centered and raw-moment formulas are mathematically equivalent, but the centered formula is often more stable numerically.

Meaning

Variance is the second central moment of the Born distribution for measuring $A$ in the specified state. It measures the distribution’s spread around its mean. It is not, by itself:

• an apparatus calibration error;
• a measurement disturbance;
• the error of an estimate of the mean;
• the width of the wavefunction in every representation;
• a complete description of the outcome distribution.

Two states can have the same expectation and variance while differing in skewness, tails, multimodality, or higher moments.

The standard deviation is the root-mean-square distance from the expectation:

$$\Delta_\rho A = \left[ \int_{\mathbb R} (\lambda-\langle A\rangle_\rho)^2 \,d\mu_\rho^A(\lambda) \right]^{1/2},$$

where $\mu_\rho^A$ is the Born probability measure of $A$.

Pure and mixed states

For a normalized pure state,

$$\langle A\rangle_\psi = \langle\psi\rvert A\lvert\psi\rangle.$$

The norm form

$$(\Delta_\psi A)^2 = \left\lVert (A-\langle A\rangle_\psi I) \lvert\psi\rangle \right\rVert^2$$

makes nonnegativity immediate:

$$(\Delta_\psi A)^2\geq0.$$

For a density operator,

$$\langle A\rangle_\rho = \operatorname{Tr}(\rho A),$$

and

$$(\Delta_\rho A)^2 = \operatorname{Tr}(\rho A^2) - \left[ \operatorname{Tr}(\rho A) \right]^2.$$

The trace formula includes pure states by setting $\rho=\lvert\psi\rangle\langle\psi\rvert$.

Discrete, continuous, and mixed spectra

If

$$A = \sum_a aP_a, \qquad p(a) = \operatorname{Tr}(\rho P_a),$$

then

$$\langle A\rangle = \sum_aap(a)$$

and

$$(\Delta A)^2 = \sum_a (a-\langle A\rangle)^2 p(a).$$

Degeneracy is already included when $P_a$ projects onto the full eigenspace. Do not sum separate probabilities for arbitrarily chosen vectors inside that eigenspace unless the measurement actually resolves them.

If the Born law has a density $p_A(\lambda)$,

$$(\Delta A)^2 = \int_{-\infty}^{\infty} (\lambda-\langle A\rangle)^2 p_A(\lambda)\,d\lambda.$$

For one-dimensional position,

$$(\Delta X)^2 = \int_{-\infty}^{\infty} (x-\langle X\rangle)^2 \lvert\psi(x)\rvert^2\,dx.$$

For a spectrum with discrete and continuous parts, include both:

$$\begin{aligned} (\Delta A)^2 ={}& \sum_n (a_n-\langle A\rangle)^2w_n\\ &+ \int_{\mathbb R} (\lambda-\langle A\rangle)^2 p_{\mathrm{ac}}(\lambda)\,d\lambda. \end{aligned}$$

Domain and existence conditions

Normalization does not guarantee a finite variance. For a pure state and self-adjoint $A$, a finite second spectral moment requires

$$\int_{\mathbb R} \lambda^2 \,d\mu_\psi^A(\lambda) <\infty.$$

Equivalently,

$$\lvert\psi\rangle \in \mathcal D(A).$$

Then $A\lvert\psi\rangle$ is a Hilbert-space vector and

$$\langle A^2\rangle_\psi = \lVert A\lvert\psi\rangle\rVert^2$$

in the spectral or quadratic-form sense. The literal vector expression $\langle\psi\rvert A^2\lvert\psi\rangle$ would demand $\lvert\psi\rangle\in\mathcal D(A^2)$, which is stronger than necessary.

For a density operator, a standard sufficient condition is

$$\operatorname{Tr}(\rho A^2)<\infty$$

with the product understood through the positive spectral calculus. Formal matrix multiplication cannot override domain or convergence failures in infinite dimensions.

If the second moment diverges, write

$$\operatorname{Var}_\rho(A)=\infty$$

when that extended value is meaningful. Do not subtract two divergent quantities in the raw-moment formula.

Zero variance and definite values

For a pure state with finite variance,

$$\Delta_\psi A=0$$

if and only if

$$(A-\langle A\rangle_\psi I) \lvert\psi\rangle=0.$$

Thus the state lies in the eigenspace with eigenvalue $\langle A\rangle_\psi$.

For a mixed state, zero variance means the support of $\rho$ lies entirely in one eigenspace of $A$. The state can still be mixed within a degenerate eigenspace. Therefore zero variance for one observable does not imply that the state is pure.

Affine changes and units

For real $\alpha$ and $\beta$,

$$\langle\alpha A+\beta I\rangle = \alpha\langle A\rangle+\beta,$$

while

$$\operatorname{Var}(\alpha A+\beta I) = \alpha^2 \operatorname{Var}(A).$$

Consequently,

$$\Delta(\alpha A+\beta I) = \lvert\alpha\rvert\Delta A.$$

Adding a constant shifts every outcome without changing the spread. Rescaling the measurement scale rescales standard deviation by the magnitude of the same factor.

The dimensions are

$$[\operatorname{Var}(A)] = [A]^2, \qquad [\Delta A]=[A].$$

This is a fast way to detect the common error of identifying variance itself with $\Delta A$.

Two-outcome shortcut

Suppose $A$ has only outcomes $a$ and $b$, with probabilities $p$ and $1-p$. Then

$$\langle A\rangle = pa+(1-p)b$$

and

$$\operatorname{Var}(A) = p(1-p)(a-b)^2.$$

The variance is maximal at $p=1/2$ and vanishes at $p=0$ or $p=1$.

For a Pauli observable

$$\sigma_{\mathbf n} = \mathbf n\cdot\boldsymbol{\sigma}, \qquad \lVert\mathbf n\rVert=1,$$

one has $\sigma_{\mathbf n}^2=I$, so

$$(\Delta\sigma_{\mathbf n})^2 = 1- \langle\sigma_{\mathbf n}\rangle^2.$$

For a spin component

$$S_{\mathbf n} = \frac{\hbar}{2}\sigma_{\mathbf n},$$ $$(\Delta S_{\mathbf n})^2 = \frac{\hbar^2}{4} \left[ 1- \langle\sigma_{\mathbf n}\rangle^2 \right].$$

Mixtures and the law of total variance

Let a preparation be described by

$$\rho = \sum_kq_k\rho_k, \qquad q_k\geq0, \qquad \sum_kq_k=1.$$

Then

$$\begin{aligned} \operatorname{Var}_\rho(A) ={}& \sum_kq_k \operatorname{Var}_{\rho_k}(A)\\ &+ \sum_kq_k \left( \langle A\rangle_{\rho_k} - \langle A\rangle_\rho \right)^2. \end{aligned}$$

The first term is the average spread within the component states. The second is the classical spread of their means. Mixing can therefore increase variance even when each component has zero variance.

This decomposition depends on the chosen ensemble representation of $\rho$, although the total on the left does not. Different ensembles for the same density operator can divide the total into different within- and between-component contributions.

Spectral bounds

If the spectrum of $A$ lies in the finite interval $[a_{\min},a_{\max}]$, then

$$0 \leq \operatorname{Var}_\rho(A) \leq \frac{ (a_{\max}-a_{\min})^2 }{4}.$$

The upper bound is reached by a distribution with equal probability at the two endpoints when the endpoint eigenspaces are available. A result outside this range signals an incorrect state, operator, probability distribution, or calculation.

Relation to uncertainty formulas

For two observables $A$ and $B$, the standard deviations enter the Robertson bound

$$\Delta A\,\Delta B \geq \frac{1}{2} \left\lvert \langle[A,B]\rangle \right\rvert.$$

The stronger Schrödinger form also contains the covariance of the centered observables. These inequalities constrain a pair of outcome distributions. They do not redefine variance, and they do not say that measuring $A$ necessarily disturbs $B$ by an amount $\Delta B$.

The derivations and equality conditions belong to General Uncertainty Relations.

Statistical estimation and detector noise

For $N$ independent measurements of the same observable in the same state, the sample mean has standard deviation

$$\Delta\overline A = \frac{\Delta A}{\sqrt N}.$$

This is the standard error of the sample mean under the independent, identically distributed model. It is distinct from the single-shot quantum spread $\Delta A$.

If independent zero-mean detector noise with variance $\sigma_{\mathrm{det}}^2$ is added to the ideal outcome, then

$$\operatorname{Var}(A_{\mathrm{obs}}) = \operatorname{Var}(A) + \sigma_{\mathrm{det}}^2.$$

Correlated, biased, state-dependent, or deconvolved noise requires a more specific measurement model. One cannot identify an observed histogram width with intrinsic quantum variance without calibration.

Numerical stability

The raw-moment formula

$$\langle A^2\rangle-\langle A\rangle^2$$

can subtract two large, nearly equal floating-point numbers. Roundoff may then produce a small negative result even though exact variance is nonnegative.

Prefer one of the following:

• evaluate the centered norm $\lVert(A-\langle A\rangle I)\lvert\psi\rangle\rVert^2$;
• sum centered outcome deviations;
• use a stable online variance algorithm for sampled data;
• symmetrize numerically Hermitian inputs;
• treat only a negative value consistent with a documented tolerance as roundoff.

A materially negative variance is not repaired by an absolute value. Diagnose normalization, Hermiticity, basis, and arithmetic first.

Assumptions

• The state is normalized and valid.
• $A$ is self-adjoint when interpreted as an observable.
• State and operator act on the same Hilbert space.
• The second spectral moment is finite.
• Discrete probabilities include every eigenspace and continuous formulas use the correct measure.
• Trace expressions with unbounded operators are defined.
• Statistical sample formulas assume stable, independent preparations unless a different model is stated.
• Detector-noise decompositions assume the specified independence and bias conditions.

Validity and limitations

Variance and standard deviation are exact properties of a Born distribution. They summarize spread but do not determine the complete distribution. They can fail to exist for normalizable states with heavy spectral tails.

The variance of a non-self-adjoint operator can be defined in several inequivalent ways. This card concerns self-adjoint observables. For general operators, expressions such as

$$\left\langle (B-\langle B\rangle)^\dagger (B-\langle B\rangle) \right\rangle$$

must be labeled explicitly rather than silently called the same variance.

Calculation checks

• Verify $\operatorname{Var}(A)\geq0$ within numerical tolerance.
• Check $[\Delta A]=[A]$ and $[\operatorname{Var}(A)]=[A]^2$.
• In an eigenstate or one-eigenspace mixture, the variance must vanish.
• For $A=\alpha I$, the variance must vanish in every state.
• Shifting $A$ by $\beta I$ must leave the variance unchanged.
• Scaling $A$ by $\alpha$ must scale variance by $\alpha^2$.
• A bounded-spectrum result must obey the range bound.
• A unitary basis change applied to both state and observable must preserve the result.
• For a two-outcome law, compare with $p(1-p)(a-b)^2$.
• When using the raw-moment form, compare with a centered calculation if cancellation is plausible.

Minimal worked uses

Spin one-half

Let

$$\lvert\psi\rangle = \alpha\lvert+\rangle_z + \beta\lvert-\rangle_z, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.$$

For

$$S_z = \frac{\hbar}{2}\sigma_z,$$ $$\langle S_z\rangle = \frac{\hbar}{2} \left( \lvert\alpha\rvert^2 - \lvert\beta\rvert^2 \right),$$

while $S_z^2=\hbar^2I/4$. Hence

$$(\Delta S_z)^2 = \hbar^2 \lvert\alpha\rvert^2 \lvert\beta\rvert^2.$$

The spread vanishes for either eigenstate and is maximal for equal outcome probabilities.

Gaussian position law

For the normalized density

$$p(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp \left[ - \frac{(x-x_0)^2}{2\sigma^2} \right],$$ $$\langle X\rangle=x_0, \qquad (\Delta X)^2=\sigma^2, \qquad \Delta X=\sigma.$$

The symbol $\sigma$ in this parameterization is the standard deviation, not the variance.

Derivation and canonical home

Variance and Standard Deviation owns the derivation from the Born measure, the pure-state norm form, finite-moment conditions, zero-variance characterization, mixture decomposition, and statistical interpretation.

Variance and Covariance owns the underlying probability identities. Correlations and Covariance develops joint quantum moments, and General Uncertainty Relations owns the operator inequalities.

Worked examples

• Expectation-Value Examples
• Spin Examples
• Gaussian Wave Packets
• Position–Momentum Uncertainty

Common mistakes

• Writing $\Delta A=\langle A^2\rangle-\langle A\rangle^2$.
• Confusing $\langle A\rangle^2$ with $\langle A^2\rangle$.
• Treating zero expectation as zero variance.
• Squaring each matrix element of $A$ instead of computing $A^2$.
• Forgetting degeneracy or a continuous part of the spectrum.
• Assuming a normalized state has a finite second moment.
• Requiring $\lvert\psi\rangle\in\mathcal D(A^2)$ when the norm form only requires $\lvert\psi\rangle\in\mathcal D(A)$.
• Calling intrinsic spread an apparatus error or measurement disturbance.
• Omitting the square when converting physical units.
• Inferring a full probability law from only its mean and variance.
• Hiding a materially negative numerical result with an absolute value.
• Assuming every ensemble decomposition of a mixed state assigns the same within-state and between-state variances.
• Applying $\Delta\overline A=\Delta A/\sqrt N$ to correlated or drifting trials without qualification.

Related formulas

• Expectation Value
• Born Rule
• Normalization
• Canonical Commutation Relations
• Uncertainty Relations
• Density-Matrix Expectation
• Bloch Vector

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chs. 1 and 3.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 4 and 9.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014, chs. 2 and 3.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. ed., Academic Press, 1980, ch. VIII.

Exercises

Exercise 1: a three-outcome distribution

An observable has outcomes $-1$, $0$, and $2$ with probabilities $1/4$, $1/2$, and $1/4$. Find its expectation, variance, and standard deviation.

Solution

The expectation is

$$\langle A\rangle = - \frac14 + 0 + \frac12 = \frac14.$$

The second moment is

$$\langle A^2\rangle = \frac14 + 0 + 1 = \frac54.$$

Therefore

$$\operatorname{Var}(A) = \frac54 - \left(\frac14\right)^2 = \frac{19}{16},$$

and

$$\Delta A = \frac{\sqrt{19}}{4}.$$

Exercise 2: mixing eigenstates

Let $\rho_1$ and $\rho_2$ be eigenstates of $A$ with eigenvalues $a_1$ and $a_2$. For

$$\rho = q\rho_1 + (1-q)\rho_2,$$

show that the total variance is entirely between components.

Solution

Each component is supported in one eigenspace, so

$$\operatorname{Var}_{\rho_1}(A) = \operatorname{Var}_{\rho_2}(A) = 0.$$

The two-outcome formula gives

$$\operatorname{Var}_\rho(A) = q(1-q)(a_1-a_2)^2.$$

This equals the variance of the component means, so the within-component term in the law of total variance vanishes.

Exercise 3: sample size

An observable has single-shot standard deviation $\Delta A=3.0$ in the prepared state. Assuming independent trials, how many measurements are needed to make the standard deviation of the sample mean at most $0.03$?

Solution

Use

$$\Delta\overline A = \frac{\Delta A}{\sqrt N}.$$

The requirement is

$$\frac{3.0}{\sqrt N} \leq 0.03,$$

so $\sqrt N\geq100$ and therefore

$$N\geq10^4.$$

This calculation assumes independent, identically distributed trials and does not include systematic detector error.