Mathematical Model

Scoring Methodology

Full audit specification of how response vectors are transformed into continuous trait and type coordinates.

1. Item Response Transformation

Each statement is presented on a fully labeled 5-point Likert scale (1 = Strongly disagree, 2 = Disagree, 3 = Neutral, 4 = Agree, 5 = Strongly agree). Let raw response be r ∈ [1, 2, 3, 4, 5].

FORWARD KEYED ITEMS:k = r

Answers directly reflect higher endorsement of the keyed trait.

REVERSE KEYED ITEMS:k = 6 - r

Inverts the response scale to maintain directional alignment.

2. Linear Scale Score (0 to 100)

For each scale containing $n$ items, the mean keyed response $M$ is computed:

M = (∑ k_i) / n

The mean (ranging from 1.0 to 5.0) is then linearly transformed onto a standard 0 to 100 scale:

Scale Score = 25 × (M - 1)

Mean 1.0 = 0

Mean 2.0 = 25

Mean 3.0 = 50

Mean 4.0 = 75

Mean 5.0 = 100

Important: Scale score, not population percentile. An 80 on Open-Mindedness means your average keyed response was 4.2 out of 5; it does not claim you score higher than 80% of human beings until a representative normative sample is applied.

3. Continuous Type Lens & Clarity Bands

The four preference axes (E/I, S/N, T/F, J/P) are calculated on a continuous 0–100 scale using an internal high-pole convention: Extraversion (100) vs Introversion (0), Intuition (100) vs Sensing (0), Thinking (100) vs Feeling (0), and Judging (100) vs Perceiving (0).

Distance from Midpoint (\|Score - 50\|)	Provisional Clarity Band	Interpretation
< 5 points	Balanced	Equator zone; easily shifts situational context.
5 to 14 points	Slight lean	Mild directional preference with notable flexibility.
15 to 29 points	Clear lean	Established cognitive default across most conditions.
≥ 30 points	Strong lean	Highly salient cognitive orientation with low ambiguity.

These bands replace pseudo-scientific “95% confidence” numbers with honest descriptive intervals until empirical standard error of measurement (SEM) data exist.

4. Deterministic Synthesis Rules

The DualLens report is not generated by a random LLM or statistical averaging. It is powered by versioned, pure deterministic rules organized into four distinct classes:

1. Reinforcement:

Evaluates when correlated dimensions confirm an underlying pattern (e.g. Introversion + low Big Five Extraversion).

2. Differentiation:

Evaluates when related scales diverge (e.g. Thinking preference + high Agreeableness).

3. Expansion:

Introduces Negative Emotionality as an independent stress-reactivity layer omitted by classic types.

4. Interaction:

Interprets how two independent traits combine into practical behavioral compounds.

5. Selective Shared-Item Architecture

DualLens maintains strict psychometric discipline while acknowledging valid construct overlap:

Pure Big Five Domains: Big Five trait scales are scored exclusively from their primary items. No type-only items influence Big Five scores.
Provisional Secondary Mappings: Type axes combine dedicated type items with explicitly supported secondary cross-scores from overlapping Big Five items (E/I with Extraversion, S/N with Openness, T/F with Agreeableness, J/P with Conscientiousness).
Independent Stress Modifiers: Negative Emotionality has zero cross-mappings into type axes, ensuring stress reactivity acts as an unconfounded cross-cutting layer.

This design guarantees that shared items reflect genuine theoretical overlap (“related, not equivalent”) without collapsing one framework into the other or compromising discriminant validity.