Mean

📐 Definition

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

$$\mu = \frac{1}{N}\sum_{i=1}^{N} x_i$$

Copied

For continuous fields $f$ over domain $\Omega$ with measure $|\Omega|$, the mean is $\mu = |\Omega|^{-1} \int_{\Omega} f(x)\,dx$.

Domain and Codomain

Applies to real or complex samples with finite first moment. Output shares the same field (complex mean taken componentwise).

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⚙️ Key Properties

Linear: mean of a linear combination equals the same combination of means. Minimizes the $L^2$ error to samples among constant estimators.

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For symmetric distributions around $0$, $\mu = 0$. As $N \to \infty$, the sample mean converges to the expectation when moments exist (law of large numbers).

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🎯 Special Cases and Limits

• Constant samples $x_i=c$ give $\mu=c$.
• For symmetric distributions about $0$, $\mu=0$.

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🔗 Related Functions

Variance measures second central moment around the mean; RMS combines mean and magnitude squared.

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Usage in Oakfield

Oakfield computes means as part of its runtime field diagnostics:

• SimFieldStats.mean_re / mean_im track the arithmetic mean of real/imag components (with mean kept as a legacy alias of mean_re).
• These statistics are computed efficiently during diagnostics updates and are used for baseline/offset monitoring and for derived metrics.

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Historical Foundations

📜 Averages and Expectation

The arithmetic mean is the canonical “average” and the finite-sample analogue of expectation in probability.

🌍 Modern Perspective

Means are ubiquitous summary statistics and also appear as minimizers of squared error among constant approximations.

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📚 References

• Grimmett & Stirzaker, Probability and Random Processes
• Cover & Thomas, Elements of Information Theory