Variance

📐 Definition

For samples $\{x_i\}_{i=1}^{N}$ with mean $\mu$,

$$\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N} |x_i - \mu|^2$$

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An unbiased estimator uses $1/(N-1)$ when treating the data as a sample from a larger population.

Domain and Codomain

Requires finite second moment; applies to real or complex data. With the modulus-squared convention above, $\sigma^2$ is always a nonnegative real number.

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⚙️ Key Properties

Translation-invariant: adding a constant to all samples leaves variance unchanged. With the modulus-squared convention, scaling obeys $\operatorname{Var}(c x) = |c|^2 \operatorname{Var}(x)$ (for real $c$ this reduces to $c^2\operatorname{Var}(x)$).

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Zero variance if all samples are equal. For Gaussian data, variance equals the squared standard deviation parameter.

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🎯 Special Cases and Limits

• Constant samples give $\sigma^2=0$.
• For zero-mean signals, $\sigma^2$ equals RMS$^2$ (when RMS is defined with $x_i^2$ or $|x_i|^2$).

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🔗 Related Functions

RMS combines variance and mean; Welford online statistics provide stable incremental updates.

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Usage in Oakfield

Oakfield computes variances as part of SimFieldStats:

• Component variances: var_re / var_im track population variance of real/imag components.
• Magnitude variance: var_abs tracks variance of $|u|$ (useful when sign/phase is less meaningful than amplitude).
• These feed thresholds/diagnostics (e.g. detecting instability, saturation, or noise growth).

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Historical Foundations

📜 Second Moments

Variance is the standard second central moment and measures typical spread around the mean.

🌍 Modern Perspective

It is a core diagnostic for uncertainty, noise, and stability monitoring and admits stable online update formulas for streaming settings.

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📚 References

• Grimmett & Stirzaker, Probability and Random Processes
• Chan, Golub, LeVeque, “Algorithms for Computing the Sample Variance” (1983)