Exact decimal interval arithmetic · one-dimensional linear model

Controlled 1D Linear Interval Stack

Propagate documented lower and upper deviations through a scalar closure equation with coefficients +1 or −1. The result is a deterministic interval enclosure, not an RSS probability, confidence level, fit decision or ISO conformity verdict.

Asymmetric deviationsExact base-10 arithmeticNo statistical assumptions

Do not substitute drawing half-tolerances into RSS and call the result “3σ” or “99.73%”.RSS combines quantities interpreted as standard deviations or standard uncertainties, with covariances where appropriate. A specification interval alone does not establish a distribution, process centering, independence or coverage probability.

Documented scalar closure equation

Each active row represents one physical length term. Enter signed lower and upper deviations relative to its nominal value.

Common unit

Drawing, loop and source record 3–500 characters

#Term labelSignNominasiLower deviationUpper deviation

1Complete every field in an active row.

2Complete every field in an active row.

3Complete every field in an active row.

4Complete every field in an active row.

5Complete every field in an active row.

6Complete every field in an active row.

7Complete every field in an active row.

8Complete every field in an active row.

I verified one scalar 1D closure equation, the +/− coefficients, common unit, datum path and that every active row refers to the same documented assembly state. I understand that this is a Cartesian interval enclosure. It does not calculate RSS probability, process yield, measurement uncertainty, assembly function, 2D/3D geometry or conformity.

Nominal closure value

—

Deterministic interval span

—

Minimum interval value

—

Maximum interval value

—

Lower deviation from nominal

—

Upper deviation from nominal

—

Evaluated chain

—

Source record

—

Exact interval model

Y = Σ s_ix_i, where s_i ∈ {+1, −1}
x_i ∈ [N_i + d_Li, N_i + d_Ui]
For s_i = +1: y_Li = N_i + d_Li, y_Ui = N_i + d_Ui
For s_i = −1: y_Li = −(N_i + d_Ui), y_Ui = −(N_i + d_Li)
Y_min = Σy_Li, Y_max = Σy_Ui

This is general interval arithmetic, not an ISO-prescribed formula. For the entered linear model it encloses every combination of independent term values inside their declared intervals. Coefficients other than ±1, nonlinear geometry and dependent/correlated term constraints are outside scope.

Why the former RSS result was removed

NIST TN 1297 describes RSS as a method for combining components estimated as standard deviations or standard uncertainties, with covariances as appropriate. A drawing tolerance half-width is not automatically a standard deviation. Consequently, √Σt² cannot be labelled “3σ”, “99.73% confidence” or expected production yield unless a validated statistical model supplies the distributions, centering, dependence/covariance and coverage interpretation.

No process capability, rejection rate or probability is calculated.
No normal distribution or independent-variable assumption is silently imposed.
No RSS output is presented beside the deterministic interval.

Scope exclusions

2D/3D vector loops, angular effects, datum mobility, geometric tolerances and contact geometry.
Thermal expansion, deformation, preload, wear, surface texture and operating-state changes unless already incorporated into controlled term limits.
Assembly fit/function, clearance acceptance, interference risk or safety decisions.
Measurement uncertainty and ISO 14253 conformity decision rules.
Statistical tolerancing, process distributions, covariance and Monte Carlo analysis.

Required engineering step: a qualified reviewer must validate the closure equation and specification operators against the complete drawing/CAD definition, datum system, operating state and controlled standards before using the interval in a design decision.

Standards lifecycle and boundary

Document	Verified public status	Kepentingan
ISO 14405-1:2025, edition 3	Published; replaces withdrawn ISO 14405-1:2016.	Indication of linear sizes and specification operators. This worksheet does not infer an operator; limits must already be controlled.
ISO 2768-2:1989	Withdrawn 2021-02-04; revised by ISO 22081:2021.	Do not present this withdrawn geometrical-general-tolerance document as current. No class table is reproduced here.
ISO 22081:2021, edition 1	Published; confirmed in 2026.	General geometrical and general size specifications under its stated scope. It does not create a universal tolerance-stack probability formula.
NIST TN 1297, section 5	Official open metrology guidance.	Clarifies that RSS combines standard-uncertainty/standard-deviation components and covariances as appropriate; it is not cited as an ISO tolerance-stack rule.

Exact normative interpretation still requires licensed controlled copies of the applicable standards and the complete product specification.

Official source records

ISO 14405-1:2025 official record — edition, status, scope and lifecycle; accessed 2026-07-15.
ISO 2768-2:1989 official withdrawn record — withdrawal and replacement; accessed 2026-07-15.
ISO 22081:2021 official record — current scope and confirmation; accessed 2026-07-15.
NIST TN 1297, section 5 — what RSS combines; accessed 2026-07-15.