Statistical Description of Signals: Mean and Variance

Why Statistics in DSP?

Most real-world signals—speech, EEG, seismic, radar—are random (non-deterministic). They cannot be described by exact formulas. Instead, we characterize them using statistical parameters that reveal their nature and guide processing choices.

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First-Order Statistics: Central Tendency

Mean (Average / DC Component)

For a discrete-time signal $x[n]$ of length $N$:

\[\mu_x = \frac{1}{N} \sum_{n=0}^{N-1} x[n]\]

Continuous-time: $\mu_x = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} x(t) dt$

Physical meaning: The DC offset. For zero-mean signals, the mean is zero.

Median

The middle value when samples are sorted. Robust to outliers unlike the mean.

Signal	Mean	Median	Best For
Gaussian noise	0	0	Symmetric distributions
Speech with silence	> 0	≈ 0	Heavy-tailed, outliers
ECG baseline wander	Drifts	Stable	Non-stationary

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Second-Order Statistics: Spread and Shape

Variance

\[\sigma_x^2 = \frac{1}{N} \sum_{n=0}^{N-1} (x[n] - \mu_x)^2 = \frac{1}{N} \sum_{n=0}^{N-1} x^2[n] - \mu_x^2\]

Physical meaning: Average power (for zero-mean signals). Units: $x^2$.

Standard Deviation

\[\sigma_x = \sqrt{\sigma_x^2}\]

Physical meaning: RMS amplitude. Same units as signal.

Comparative Example: Same Mean, Different Standard Deviations

Signal A: {9, 10, 11}    μ = 10, σ = 0.82  (tight cluster)
Signal B: {0, 10, 20}    μ = 10, σ = 8.16  (wide spread)

Same center, vastly different “energy” and variability.

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