Quasi-arithmetic mean¶
Average values by mapping them through a continuous strictly monotone generator, taking an arithmetic mean in transformed coordinates, and mapping the result back.
Core Idea¶
For a nondegenerate real interval \(I\), a continuous strictly monotone generator \(f:I\to\mathbb R\), and inputs \(x_1,\ldots,x_n\in I\), the quasi-arithmetic mean generated by \(f\) is \(M_f(x_1,\ldots,x_n)=f^{-1}\!\left(n^{-1}\sum_{i=1}^{n}f(x_i)\right)\). Strict monotonicity makes the inverse well defined on \(f(I)\), while continuity and the interval hypothesis ensure that averaging transformed values stays in the range. Affine changes \(g=af+b\), with \(a\ne0\), generate the same mean, so the generator is a coordinate choice rather than unique data.
Scope of Application¶
The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Quasi-arithmetic mean itself, not metaphors based only on resemblance.
- Theory of means. Comparing mean families by invariance, monotonicity, and functional equations.
- Measurement scales. Choosing an averaging coordinate compatible with how a quantity is represented.
- Statistics. Expressing transformed-location summaries while retaining the original reporting scale.
- Decision aggregation. Combining commensurable values under declared separability and transformation assumptions.
- Information aggregation. Studying decomposable summaries generated by a scalar coordinate change.
- Inequality theory. Ordering different generators and locating equality cases under adequate smoothness.
Clarity¶
A clear account of Quasi-arithmetic mean must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. Declare the interval, generator, monotonic direction, and whether weights are equal. Check the generator's inverse on the entire transformed average range. Treat affine-equivalent generators as the same mean rather than different catalog identities. Name domain exclusions before using logarithmic, reciprocal, or power special cases.
Manages Complexity¶
Quasi-arithmetic mean manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: input interval supplies a common ordered domain makes transformed averaging and inversion meaningful.; generator supplies a continuous strictly monotone function selects the coordinate in which arithmetic averaging occurs.; transformed observations supplies each input is mapped through the same generator before transformed-coordinate averaging.; equal-weight average supplies the arithmetic mean of transformed values, combining multiple positions only when n >= 2.; inverse transformation supplies the summary returns to the original input scale..
Abstract Reasoning¶
- Verify that all inputs lie in one nondegenerate interval on which the proposed generator is continuous and strictly monotone.
- Map every input through the same generator and preserve the input multiplicity or declared weights.
- Compute the arithmetic average in transformed coordinates.
- Apply the inverse generator and verify that the result returns to the original interval.
- Test idempotency by setting every input equal and symmetry by permuting the inputs.
Knowledge Transfer¶
For evaluations with at least two input positions, Aggregation is the conditional upward abstraction: transformed-coordinate averaging collapses multiple scalar inputs to one summary. For n = 1, the formula returns its sole input, so the Aggregation relation does not apply. Within theory of means, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the applicable parent-level skeleton should travel. Reusing the label Quasi-arithmetic mean after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. Recognize the domain-specific pattern first, then lift the parent relation only when its multi-input condition holds.
Relationships to Other Abstractions¶
Current abstraction Quasi-arithmetic mean Domain-specific
Parents (1) — outgoing typed relations
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Quasi-arithmetic mean is a kind of, conditional Aggregation Prime
For n >= 2, transformed-coordinate averaging collapses multiple scalar input positions to one summary; the n = 1 evaluation is an identity, not Aggregation.
Condition / exception Evaluations with at least two input positions (n >= 2); at n = 1 the mean is the input itself and no many-item collapse occurs.
Children (1) — abstractions linked here by typed relations
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Harmonic Mean Domain-specific is a kind of Quasi-arithmetic mean
Every positive equal-weight harmonic mean is the quasi-arithmetic mean generated by the reciprocal function; the broader family also contains nonreciprocal generators.
Hierarchy path (1) — routes to 1 parentless root
- Quasi-arithmetic mean → is a kind of, conditional Aggregation → is a kind of Transformation → is a kind of, typical Function (Mapping)
Neighborhood in Abstraction Space¶
Quasi-arithmetic mean sits in a sparse region of the domain-specific corpus (95th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2769 abstractions)
Nearest neighbors
- Jensen's Inequality — 0.79
- Neuman–Sándor Mean — 0.78
- Monotonic Function — 0.78
- Carleman's equation — 0.77
- Condition Number — 0.77
Computed from structural-signature embeddings · 2026-10-25