Data & measurement
Continuity metrics emphasize intergenerational transmission, corpus completeness, and structural stability. Keep cohort definitions fixed across measurement waves. 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: Measures the stability of grammatical structure over time. 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. Word problem — pilot estimates
Word problem
Evaluate “Structural Stability” 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 “Structural Stability” 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.
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 “Structural Stability” 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.
Source
ALL_FORMULAS.md — Continuity