Technology / Technical guide
Nonlinear Averages: Why Cubing the Mean Changes the Result
Applying a nonlinear relationship before averaging generally differs from applying it to an average input. For wind-speed mathematics, a cube highlights that difference, but it does not supply a real turbine power curve. The order of operations, domain and weights must remain attached to the quantity being calculated.
Averaging and transformation need not commute.
Compare the two calculations directly, then examine which mathematical conditions establish an inequality rather than an electrical yield.
01 / Order of operations
A fixed affine relation is a special case.
For nonnegative weights w_i summing to one, μ = Σw_i x_i defines a weighted mean.
For nonnegative weights w_i summing to one, μ = Σw_i x_i defines a weighted mean. A fixed affine relation f(x) = ax+b gives Σw_i f(x_i) = f(μ). A nonlinear f does not generally share that identity.
The weights describe the represented quantity. Duration weights belong to a time average, while equal observation weights give a sample average. Changing those weights changes both calculations; it is not merely a different display format for the same result.
02 / Time example
Three invented speeds give different cubic statistics.
Hypothetical equal-duration values: 3, 5 and 7 m/s have mean 5 m/s.
Hypothetical equal-duration values: 3, 5 and 7 m/s have mean 5 m/s. Cubing that mean gives 125 (m/s)³. Averaging their cubes gives (27+125+343)/3 = 165 (m/s)³.
The difference is 40 (m/s)³. This is a temporal arithmetic example, not a rotor-area average or a measurement. It establishes no percentage increase in electrical production and does not describe any MARSTON site or model.
03 / Convexity
The inequality requires the represented domain.
On nonnegative values, f(x) = x³ is convex, so Jensen’s inequality gives f(μ) ≤ Σw_i f(x_i), with existing finite quantities and the stated weights.
On nonnegative values, f(x) = x³ is convex, so Jensen’s inequality gives f(μ) ≤ Σw_i f(x_i), with existing finite quantities and the stated weights. Boyd and Vandenberghe explain this convex-function relationship.
A signed velocity component can be negative, and x³ is not convex over the whole real line. The nonnegative-speed inequality therefore cannot be copied unchanged to every signed signal. A concave function reverses the corresponding comparison under its own domain conditions.
04 / Higher moments
Variance and asymmetry enter the cubic mean.
Let deviations be d_i = x_i−μ, with weighted mean deviation zero.
Let deviations be d_i = x_i−μ, with weighted mean deviation zero. Expanding (μ+d_i)³ gives the identity Σw_i x_i³ = μ³+3μσ²+m3, where σ² = Σw_i d_i² and m3 = Σw_i d_i³.
For the invented three values, σ² = 8/3 (m/s)² and m3 = 0, so the extra term is 3×5×8/3 = 40 (m/s)³. Equal means alone cannot identify those higher moments; their units also differ from the mean.
05 / Other factors
A product average can retain shared variation.
If a represented relationship also contains changing density ρ, averaging ρx³ is not generally the same as multiplying the mean density by the mean cube.
If a represented relationship also contains changing density ρ, averaging ρx³ is not generally the same as multiplying the mean density by the mean cube. Their difference is Cov(ρ,x³) when those weighted moments exist.
That algebra concerns the represented factors, not an assumed turbine performance law. Likewise, measurement noise passing through a nonlinear transformation can alter its average; a zero-mean error does not automatically imply zero transformed bias.
06 / Scope
A real output relationship has its own shape.
Actual electrical output can involve changing coefficients, operating states, conversion losses and output limits.
Actual electrical output can involve changing coefficients, operating states, conversion losses and output limits. Its relationship need not be globally convex or simply proportional to a cube. Applying Jensen’s inequality requires the actual function on the represented domain.
The distribution guide summarises resource records, and the power-curve guide explains output evidence. This page supplies mathematical identities, not a yield estimate, control law or product performance claim.
Related technical guides: Wind Air Density: Temperature, Pressure & Humidity · Rotor Spatial Averaging: Point Wind, Area Mean & Equivalent Speed · Converter Instantaneous Power, Storage & Dissipation.
General background: Boyd and Vandenberghe: Convex functions and Jensen’s inequality. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| μ | Σw_i x_i | Weights are nonnegative and sum to one. |
| Average cube | Σw_i x_i³ | Different from μ³ in general. |
| σ² | Weighted second central moment | Units are squared input units. |
| m3 | Weighted third central moment | Retains signed asymmetry and cubic units. |
| Jensen comparison | Convex function on the actual domain | Not a universal electrical-generation inequality. |
Engineering questions.
Can I generally transform only the average input?
Not when the intended result is the average of a nonlinear output.
Does a fixed affine relationship commute with averaging?
Yes, under the stated common normalised weights.
Is the cubic inequality valid for all signed values?
No. Its convexity domain must be retained.
Does the mean determine the mean cube?
No. Variance and the third central moment also contribute.
Does the 165 value estimate generation?
No. Its units and purpose are mathematical speed-cube statistics.
Is changing density independent of speed automatically?
No. Shared variation can remain through their covariance.
Technical enquiry
Discuss the technical question.
Applying a nonlinear relationship before averaging generally differs from applying it to an average input. Share the model and applicable operating or design information for an engineering discussion.