Domain: Natural Science — Statistics & Data Science
Target concept: Heteroscedasticity: The points in the scatter plot exhibit a fan-shaped dispersion pattern as the independent variable increases.

Video generation prompt: Create a two-dimensional scatter plot without points, where the x-axis represents the independent variable and the y-axis represents the dependent variable, with the independent variable increasing from left to right. Gradually add data points with heteroscedasticity to the plot. The plotting process is demonstrated in a step-by-step manner.

Evaluation guide:
(i) Target concept: Heteroscedasticity as non-constant variance in y conditional on x.
Minimum mechanism: Points must be spatially grouped by x-interval, and the y-range (max−min) of each group must strictly increase with x-position; no interpolation or smoothing: raw point positions must directly support the fan-shape inference.
(ii)Storyline: [Step 0] Empty coordinate system with labeled x-axis (left→right) and y-axis, fixed viewpoint. [Step 1] First point group appears exclusively in left x-region, narrow y-range. [Step 2] Second point group appears exclusively in middle x-region, wider y-range than Step 1. [Step 3] Third point group appears exclusively in right x-region, widest y-range. Causal transitions: Each step’s point placement directly causes increased perceived vertical dispersion at higher x. Visibility constraints: All points must remain fully visible (no occlusion), axes must stay in-frame, and x-axis orientation must be unambiguously left-to-right throughout.

Manual:
Experimental Procedure
1. Display blank 2D coordinate system with horizontal x-axis (labeled left→right) and vertical y-axis.
2. Add first group of points near left end of x-axis, tightly clustered vertically.
3. Add second group further right, with noticeably wider vertical spread.
4. Add third group near right end, with widest vertical spread — forming fan shape.
Observation of Experimental Phenomena
Scatter plot starts empty; points appear sequentially left-to-right; vertical spread of each group increases progressively; final plot shows clear fan-shaped dispersion pattern.
Summary of Experimental Principles
Heteroscedasticity: The variance of the dependent variable (y) increases as the independent variable (x) increases — visualized as widening vertical scatter along x-axis.

Prompt Grounding rubric:
level_1: No coordinate system appears at any point; no x-axis or y-axis is drawn; no data points are ever added to the frame; the clip shows unrelated content instead of a plot.
level_2: A coordinate system with x- and y-axes is drawn, but the axes carry no orientation cue (nothing indicates that x increases from left to right); zero data points are ever added to the canvas.
level_3: An empty two-dimensional plotting canvas appears with an x-axis oriented so values increase left to right, and a y-axis; starting from zero points, a single batch of data points appears all at once rather than being built up gradually across separate stages.
level_4: An empty two-dimensional plotting canvas appears with a clearly oriented x-axis (left-to-right increase) and a y-axis; points are added across at least two separate temporal stages rather than all appearing at once; every point that appears lands within the plot's axis bounds; however, the point markers are not fully uniform across stages (size, color, or shape shifts between stages), or an axis label goes missing partway through the clip.
level_5: An empty two-dimensional plotting canvas appears with a clearly labeled, oriented x-axis (left-to-right increase) and y-axis; points are added across at least three visibly separate temporal stages, with each stage adding new points only and never removing or repositioning existing ones; every point stays within the plot's axis bounds at all times; all points across every stage share identical marker size, shape, color, and opacity.

Scientific and Causal Correctness rubric:
level_1: The points show no relationship between x-position and vertical spread (they form a uniform band, a single tight cluster, a straight line, or a random cloud with constant scatter at every x-value), or no points appear at all.
level_2: A coordinate system appears, but points are added with no regard to x-position; or the vertical spread of the points stays constant as x increases; or the pattern is inverted, with points clustering tightly on the right and spreading widely on the left.
level_3: Points are added progressively from left to right along the x-axis, and the vertical spread of each new group is generally larger than the group before it; however, the pattern breaks in at least one place; for example, two neighboring groups show inconsistent spread relative to each other, or an earlier, more-leftward group's vertical range extends beyond that of a later, more-rightward group.
level_4: Points are added progressively from left to right; at each successive x-region, the vertical spread (the distance from the topmost to the bottommost point in that region) is visibly larger than in the region before it, with no region breaking this widening trend; however, either the x-axis scale is distorted enough to skew the perceived spread, or points meant for the same x-region are horizontally scattered enough to blur which region they belong to.
level_5: For any left-hand x-region compared with any x-region further to the right, the vertical range (topmost point minus bottommost point) in the right-hand region is clearly larger than in the left-hand region, holding across every region added; each stage's points occupy their own x-range without overlapping the x-range of stages added before or after it; no point drifts outside the x-region it was originally placed in; the widening, fan-shaped scatter is unambiguous in the final frame.

Spatiotemporal Consistency rubric:
level_1: The axes teleport or resize abruptly; points appear or disappear with no transition; there is a major flicker, jump-cut, or dropped frame; or the coordinate system rotates, warps, or changes perspective mid-clip.
level_2: Axis labels fade in and out inconsistently; points pop into existence with no fade-in or slide-in motion; the background lighting shifts erratically between stages; or the plot area resizes slightly between stages.
level_3: The axes and plot canvas stay fixed in position and scale for the entire clip; points only ever appear via a fade-in or slide-in motion, never an instant pop-in; there is no warping or tearing of the canvas at any point; however, the point-entry motion looks slightly abrupt (closer to a hard cut than a smooth fade), or the axis lines flicker faintly, or the background brightness shifts subtly between stages.
level_4: The camera viewpoint and coordinate system stay completely stable; every point in a given stage enters using the same smooth motion (for example, a uniform fade-in or a consistent slide-in from below); there is no lighting, shadow, or perspective change at any point; there is no stutter or discontinuity between frames; the only remaining flaw is that within one stage, the points do not all appear at exactly the same moment (a slight, visible stagger is noticeable between the first and last point of that stage), or the axis tick marks briefly shimmer during a transition.
level_5: The camera viewpoint never moves, and the plot canvas shows no perceptible drift, resize, or jitter across the entire clip; every point within a given stage appears to enter together, with no visible stagger between them, using the same smooth fade-in or slide-in motion; there is no flicker, jerk, or blur at any point transition; every stage-to-stage transition is visually continuous, with no jump cuts.