#06 Inverted variables (e.g. TBP)

Data & measurement

Normalize impact variables from heterogeneous raw measures to [0,1]. State targets v_i^target and level ceilings alongside every reported norm. Write explicit operational definitions for each symbol in your protocol, even when abbreviations look standard. Log instrument versions, sample frames, and cleaning rules whenever estimates are refreshed so longitudinal comparisons stay valid.

Solution & proof

Conceptual summary: When lower raw values mean better outcomes (e.g. Transmission Break Point), subtract from 1 so higher is always better. Treat this as a measurement recipe: map each symbol to an empirical quantity, substitute estimates, and simplify with ordinary algebra (including logarithms, min/max caps, or piecewise branches where shown). Where limits or integrals appear, approximate with discrete sums on cohorts or time steps when closed forms are impractical. Interpret the result against thresholds in the cited source and report uncertainty on inputs.

Examples

1. 1. Word problem — pilot estimates

   Word problem

   Evaluate “Inverted variables (e.g. TBP)” using the provisional estimates in the table. Compute the outcome with the formula at the top of this page.

   Provisional estimates for substitution (illustrative)

   Item	Estimate
   Primary input A	0.62
   Primary input B	0.78
   Scaling / time factor	1.0

   Solution

   Step 1 — From the table, assign a = 0.62; b = 0.78; T = 1.

   Step 2 — Substitute into the definition of “Inverted variables (e.g. TBP)” at the top of the page (same structure as the source formula).

   Step 3 — Finish any remaining algebra (sums, caps, logs, or limits) by hand or in a CAS when the expression still contains Σ, integrals, or non-numeric parameters. Tie final numbers back to your corpus or community records.

   $$\text{Given: }a=0.62,b=0.78,T=1$$

   $${\mathrm{norm}}_i=1-v_i$$

2. 2. Check — revised primary input

   Word problem

   The same measure applies with a revised first input when a late survey batch arrives (see table).

   Updated provisional estimates (illustrative)

   Item	Estimate
   Primary input A	a	0.71
   Primary input B	b	0.78
   Scaling / time factor	T	1.0

   Solution

   Step 1 — From the table, assign a = 0.71; b = 0.78; T = 1.

   Step 2 — Substitute into the definition of “Inverted variables (e.g. TBP)” at the top of the page (same structure as the source formula).

   Step 3 — Finish any remaining algebra (sums, caps, logs, or limits) by hand or in a CAS when the expression still contains Σ, integrals, or non-numeric parameters. Tie final numbers back to your corpus or community records.

   $$\text{Given: }a=0.71,b=0.78,T=1$$

   $${\mathrm{norm}}_i=1-v_i$$

Source

Impact measurement framework — Normalization Rules

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