# Definitions, scales, precision and denominators

## Selected retrospective sources

The 2018 Volume II gender workbook (Table II.B1.7.28/.33/.38) supplies exact 2009 boys/girls country means and SEs on the published retrospective score scales. The 2018 immigrant workbook (II.B1.9.9/.10) supplies 2009 classified-background composition and reading means. Original 2009 annexes provide independent corroboration. The retrospective tables, not rounded PDF labels, supply candidate values. Workbook floating-point precision is preserved in CSV with 17 significant digits; this preserves source precision rather than inventing statistical precision.

The extraction preserves original country labels, group headers, source cell and SE cell, nonnumeric estimate/SE/coverage codes, file hashes, table/version text, and bold formatting where available in XLSX. `workbook_cells.csv` retains every nonempty cell of the inspected sheets, including notes and headers. The original XLS bytes remain available where old formatting is not mapped by the XLS-to-cell reader.

The 2009 gender PDF values 478/522 reading, 508/505 mathematics and 496/495 science are **rounding checks only**. Likewise 504/423/417 for non-immigrant/combined/first-generation reading and 97.6/2.4/0.4/1.9% composition. No printed integer has been stored as an invented unrounded estimate.

## Trend scales and mode

OECD's PISA 2009 Volume V, Chapter 1, printed/PDF p.26, identifies full-domain trend baselines: reading 2000, mathematics 2003, science 2006. It explicitly excludes 2000 mathematics and 2000/2003 science from the corresponding overall-domain trends. The availability matrix therefore distinguishes a published earlier limited-domain observation from a validated trend point. Country participation, coverage and assessment changes require specific checks, not a blanket rejection of old paper-based tests.

The 2018 retrospective gender and immigrant reading tables explicitly report 2009 and 2018 and their cross-cycle comparisons, spanning the change to computer-based assessment. Their overlapping 2015/2018 cells reconcile to the current frozen series within 0.0001. This supports using those published historical points; it does not remove uncertainty due to linking or later sampling/coverage qualifications. The existing current-series source strings and later observations are retained, including their original qualifications.

For Austria, the 2009 boycott/assessment-motivation issue is a **specific** trend boundary (Volume I Annex A4, printed p.186 / PDF index 187). It explains the retrospective `m` cells; an original cross-sectional estimate is not a comparable substitute.

## Immigrant status

The modern grouping uses student and parental **birthplace**, not nationality. Non-immigrant students have at least one parent born in the assessment country; second generation are locally born with both parents born abroad; first generation are foreign-born with both parents born abroad. Combined immigrant means are extracted from the explicitly published combined column. They are not averages of generation means, and suppressed cells are never reverse-engineered.

The 2009/2018 retrospective tables and frozen 2022/2025 tables use this classification. Numeric 2009 non-immigrant, combined and first-generation reading cells coexist with `c` for second generation. In 2012, second-generation scores are `c` in all three subjects; first-generation and combined groups remain numeric. The candidate history uses only the two published combined categories. The current 2025 generation snapshot is unchanged.

The 2003 Table 4.2f uses different labels: its “first-generation” refers to locally born children of foreign-born parents (modern second generation), while “non-native” refers to foreign-born children (modern first generation). Its native heading also specifies the student's own birthplace. This needs an explicit returning-student/denominator reconciliation before numerical reuse; relabeling alone is insufficient. Published 2006 native and generation tables are located but not normalized into candidate points. Suppression in separate generations is not evidence that a combined mean never existed.

## Composition and missing information

Composition percentages are conditional on **classifiable immigrant status**. For 2009, II.B1.9.9 Iceland row 30 yields:

| Group | Exact percentage | Exact SE | Cells |
|---|---:|---:|---|
| Non-immigrant | 97.63926156483167 | 0.2498538424709436 | B30/C30 |
| Combined immigrant | 2.360738435168355 | 0.24985384247093928 | E30/F30 |
| Second generation | 0.4215747555234473 | 0.11426396199855388 | H30/I30 |
| First generation | 1.9391636796449 | 0.20912567756315775 | K30/L30 |

Non-immigrant plus combined sums to 100 within floating-point precision; first plus second equals the combined share. The rounded PDF components need not add perfectly. The exact 2018 overlap in the newer composition source confirms the continued grouping and denominator. These identities do not reveal how many pupils lack background information.

Missing background uses the distinct source reporting-sample denominator **including unknown background**, not the classified denominator. The existing note in `reference_inputs/immigrant_missing_treatment.md` cautions that the published table does not provide every eligibility/weighting rule or prove how all unknown students enter the national overall score. Those uncertainties are preserved. The original 2015–2025 missing frequencies and latest 8.5% versus 4.2% callout remain unchanged. Neither a pre-2015 zero nor a claim about national-mean inclusion is inferred.

The combined composition share in 2000 is published retrospectively in V.4.4; separate 2000 generation shares were not recovered. It is supporting availability evidence, not an instruction to extend the selected composition candidate beyond 2009.

## Uncertainty in new changes

Candidate annotations subtract endpoint means and are descriptive. The old and new endpoints are not treated as independent samples with a generic change SE. No synthetic confidence intervals, cross-cycle significance flags or suppressed estimates are produced. Country-average SE reconstruction is documented separately and does not supply the missing cross-cycle linking/covariance information.