Domain: Natural Science — Statistics & Data Science
Target concept: Partial Correlation: A scatter plot showing a distinct unidirectional distribution after controlling for variables

Video generation prompt: First, the original scatter plot of X versus Y is presented. Then, Z is introduced as a control variable. A new scatter plot of X versus Y is generated after applying statistical control methods to account for the influence of Z. The focus is on demonstrating the changes in the distribution characteristics of the scatter plot of X versus Y before and after controlling for the Z variable.

Evaluation guide:
Target concept: Partial correlation as a statistical control technique that isolates the direct X–Y relationship by removing variance shared with Z. Minimum mechanistic assumptions: (i) X, Y, Z are continuous numeric variables; (ii) Z is a confounder introducing spurious association between X and Y (this is distinct from Z being a mediator of a genuine causal path; partialling out a mediator would instead strip away a real indirect effect, not reveal a hidden one); (iii) linear regression is appropriate for residualization. Phased storyline: (1) Baseline: X–Y scatter shows weak/no clear trend due to Z-induced noise; (2) Intervention: Z is declared and statistically partialled out; (3) Outcome: X–Y scatter transforms into a narrow, oriented band, confirming existence of conditional linear dependence. Visibility constraints: Must preserve identical axis scaling and spatial registration across frames; Z must be visibly anchored (not just voiced); unidirectionality must be discernible within roughly 3 seconds of the controlled plot's onset without needing extra annotation.

Manual:
I. Basic Demonstration Information
Demonstration Title: Partial Correlation Visualization (X vs Y Before and After Controlling for Z)
Subject: Natural Science - Statistics & Data Science - Multivariate Relationships
Demonstration Purpose: To visually demonstrate how controlling for a third variable Z reveals the true bivariate relationship between X and Y by removing spurious covariance; specifically, to show the transformation from a diffuse, weakly structured X–Y scatter plot to a distinct unidirectional (linear) distribution after statistical control.

II. Demonstration Carrier/Equipment
Core Carrier: Animated dual-scatter-plot visualization (SVG/HTML5 canvas or equivalent)
Auxiliary Elements: Consistent Cartesian coordinate system (identical axis limits, scales, grid), labeled axes (X horizontal, Y vertical), on-screen textual annotations (‘Raw: X vs Y’, ‘Controlled: X vs Y | Z’), Z variable identifier (e.g., ‘Z = [contextual name]’) in persistent corner badge.

III. Demonstration Content Description
Layout: Full-frame centered scatter plots occupying ≥80% of screen width/height; white background with light gray grid; axis labels in 16pt sans-serif; point radius = 4px; raw points in medium-gray (#888); controlled points in vivid cobalt blue (#0057b7).
State Control: Both plots use identical x- and y-axis limits; no zooming, panning, or rescaling; transition occurs via smooth 0.4s crossfade with simultaneous point repositioning (no teleportation).
Comparison Baseline: Raw plot displays moderate scatter with low apparent correlation (visual r ≈ 0.3); controlled plot displays tight unidirectional linear trend (visual r ≥ 0.75), elongated along 45°–135° diagonal.

IV. Observation of Core Phenomena
Raw Plot Characteristics: Points form a broad, near-circular or mildly elliptical cloud; no dominant orientation; density uniform across quadrants; no visible stratification by Z.
Controlled Plot Characteristics: Points concentrate tightly along a single straight band (slope ≠ 0); minimal perpendicular scatter; clear head-to-tail directionality (left-low→right-high OR left-high→right-low); Z-stratified clusters (if present in raw) are eliminated.
Characteristic Stability: Axis alignment, font, color scheme, and point sizing remain invariant before/during/after transition; no flicker, jitter, or layout shift.
Auxiliary Comparison: Z badge remains fixed in top-right corner throughout; its appearance coincides precisely with onset of controlled plot; no other variables or distractors introduced.

V. Summary of Key Knowledge Points and Principles
Core Concept Definition: Partial correlation measures the strength/direction of the linear relationship between X and Y *after removing the linear effects of Z*; it isolates the unique covariance not shared with Z.
Statistical Principle: Controlling for Z involves regressing both X and Y on Z separately, then correlating their residuals — this eliminates Z-driven spurious associations and exposes latent X–Y dependence.
Visual Recognition Logic: A ‘distinct unidirectional distribution’ is operationally defined as: (i) principal axis length ≥ 4× orthogonal axis length; (ii) ≥85% of points lie within 15° of regression line; (iii) no visible sub-clusters or Z-aligned stripes remaining.

Prompt Grounding rubric:
level_1: No scatter plots are shown, or only a single static X–Y plot appears with no before/after comparison; axis labels for X, Y, or Z are absent; there is no visual cue that a control step occurred; the video shows unrelated imagery (e.g., bar charts, a single histogram, or animated shapes with no Cartesian axes).
level_2: Two scatter plots of X vs Y appear, but they are not shown in clear temporal sequence (e.g., both displayed at once rather than one after another); Z is never named or labeled on screen; there is no visual marker (title change, on-screen text, transition) distinguishing the 'before controlling for Z' plot from the 'after controlling for Z' plot; axis labels are missing or inconsistent between the two plots.
level_3: Two scatter plots of X vs Y are shown one after the other: the first is the raw/uncontrolled plot, the second follows after Z is mentioned; both plots have legible X and Y axis labels; Z is referenced only in a caption or voiceover, without any on-screen badge, icon, or annotation tied to it; the difference in point spread between the two plots is ambiguous: a viewer cannot confidently say the second plot looks more directional than the first.
level_4: Two scatter plots of X vs Y appear in clear temporal order, each with an on-screen label ('X vs Y, raw' then 'X vs Y, after controlling for Z'); the first plot shows a visibly diffuse, roughly circular or scattered cloud of points; the second plot shows a visibly narrower, elongated band of points oriented along a clear diagonal; Z appears as on-screen text (e.g., 'Controlling for Z') at the transition; fonts, colors, and axis ranges are consistent between the two plots, with no obvious mislabeling or swapped axes.
level_5: The raw X–Y scatter plot is shown first, clearly labeled, with points in one consistent color, displaying a visibly diffuse, non-directional cloud; a brief on-screen cue (e.g., 'Controlling for Z...') marks the transition; the second X–Y scatter plot uses the same axis scale and range as the first, is rendered in a distinctly different point color, and clearly shows a narrow, elongated band of points running along one consistent diagonal direction, visibly tighter and more directional than the first plot; Z is named on screen throughout the second plot (e.g., a small 'Z = [variable]' label); there is no axis rescaling, no relabeling, and no jump cut between the two plots.

Scientific and Causal Correctness rubric:
level_1: The statistical pattern shown is implausible for partial correlation: e.g., the controlled X–Y plot appears more scattered than the raw plot, or the raw plot shows a near-perfect line while the controlled plot looks random; there is no implied link between introducing Z and the change in the X–Y pattern; axis conventions are violated (e.g., Y plotted on the horizontal axis).
level_2: The controlled plot shows some change in shape (e.g., a slight tilt), but the direction of change contradicts how partialling out a confounder should behave (e.g., the raw plot trends upward and the controlled plot bends downward for no stated reason); Z is introduced but shown as unrelated to X and Y (e.g., Z appears constant or with no visual link to either variable); the transformation does not visually connect to Z's role as a confounder.
level_3: The controlled plot is visibly more linear/directional than the raw plot, consistent with removing a confounder's shared variance, but the change is underwhelming (only a slight tightening, no clearly dominant direction); Z is named on screen but its statistical role is not conveyed (no reference to 'holding Z constant,' 'controlling for Z,' or similar); there is no scientific error, but the mechanism is not made clear.
level_4: The raw X–Y plot shows a weak apparent trend (visual correlation roughly r ≈ 0.2–0.4, a loose, near-circular cloud); the controlled plot shows a markedly stronger, narrower directional band (visual correlation roughly r ≈ 0.7–0.8); on-screen text or an icon links Z to the earlier weak trend (e.g., 'Z was masking the true X–Y relationship'); the transformation is consistent with partial correlation logic: removing variance shared with confounder Z reveals a clearer X–Y relationship; there is no implied reversal of causality or claim that Z directly causes X or Y.
level_5: The full mechanism of partial correlation is shown: the raw X–Y cloud shows a diagonal trend overlaid with extra spread attributable to Z (e.g., points appear to cluster into subgroups by Z-level before control); after Z is partialled out, that subgroup-like spread disappears and only one coherent linear trend remains, with visual correlation increasing from roughly r ≈ 0.3 in the raw plot to r ≈ 0.8 or higher in the controlled plot; the transition visually reinforces that removing variance shared with confounder Z is what reveals the direct relationship, without any suggestion that Z itself directly causes X or Y; the two plots use the same axis scaling and no points are added or removed between them.

Spatiotemporal Consistency rubric:
level_1: The two scatter plots appear as a disjointed cut (no transition, mismatched background, font, or plot size); the temporal order is violated (the controlled plot appears before the raw plot); the two plots are laid out inconsistently (e.g., one left-aligned, one right-aligned, with different margins); axis labels jump position or change scale erratically between the two plots.
level_2: The two plots appear in the correct order but with inconsistent framing (different axis ranges, rotated labels, or noticeably different point sizes between the two plots); the transition between them is an abrupt cut with a noticeable blank gap rather than any visual bridge; the Z label is only spoken in voiceover and never appears on screen; the background changes abruptly between the two plots (e.g., plain white for the first, gridded or gradient for the second).
level_3: Both plots share the same canvas size and axis limits, but minor flaws are visible: the transition between plots is a quick, slightly jittery fade; the Z label appears noticeably after the second plot has already appeared rather than alongside it; point shading is inconsistent between the two plots; the first plot is held only briefly while the second lingers much longer (or vice versa), giving an uneven, rushed feel to the sequence.
level_4: The transition between the two plots is a smooth, brief crossfade with no visible cut; the coordinate system (tick marks, font size, gridline style, point size) stays identical between the two plots; the Z label appears at essentially the same moment the second plot appears, without a noticeable delay; the plot area, axis labels, and Z label all stay in the same on-screen positions throughout, with no rescaling, rotation, or cropping; the first and second plots are each held on screen long enough to be read comfortably, with the transition clearly shorter than either plot's viewing time.
level_5: The spatial and temporal presentation is seamless: the two plots are drawn on a visually matching canvas (same background, gridlines, font, and point size) with no visible shift in axis position between them; the transition is a smooth, continuous repositioning of points toward the new trend direction rather than a sudden swap; the Z label appears in sync with the second plot's arrival with no perceptible lag; gridlines, tick marks, and axis labels do not shift, jitter, or flicker at any point; the overall pacing feels natural and unhurried, with no visible layout recalibration or stutter between the two plots.