Domain: Natural Science — Statistics & Data Science
Target concept: Chi-square distribution

Video generation prompt: A white background displays the probability density function of the chi-squared distribution with degrees of freedom df = 1. Then, keeping the same scale, the degrees of freedom df are incrementally increased by one up to 5, and the changes in the probability density function are observed.

Evaluation guide:
(i) Target concept: Visual parametric evolution of chi-squared PDFs under varying degrees of freedom on a shared scale.
Minimum mechanism: The video must encode the mathematical definition f(x; k) = (1/(2^{k/2}Γ(k/2))) x^{k/2−1} e^{−x/2} for x ≥ 0, where k = df ∈ {1,2,3,4,5}, implying strict support on [0,∞), right-skew decay, and mode progression.
(ii)Storyline: Frame 0–1: White background established, static, full-frame. Frame 2–3: df=1 curve appears (steep rise at origin, rapid decay). Frame 4–5: df=2 curve overlays, visibly less skewed, smoother decay. Frame 6–7: df=3 curve overlays, peak right-shifted, moderate skew. Frame 8–9: df=4 curve overlays, further right-shifted peak, lower height. Frame 10: df=5 curve overlays, most symmetric, broadest spread. All curves coexist in final frame. Fixed frontal 2D plot perspective maintained; no camera movement, zoom, or rotation. Labels “df=1” through “df=5” remain anchored and legible beside respective curves throughout.

Manual:
Experimental Procedure
1. Display white background.
2. Plot chi-squared PDF curve for df = 1.
3. Without changing scale, overlay curve for df = 2.
4. Continue overlaying curves for df = 3, 4, 5 in ascending order.
Observation of Experimental Phenomena
- df = 1 curve rises sharply near x = 0 and is highly right-skewed.
- As df increases, the peak shifts right (mode = df − 2), height decreases, and skew reduces.
- df = 5 curve appears most symmetric and bell-like among the five.
Summary of Experimental Principles
The chi-squared distribution is a continuous probability distribution defined for x ≥ 0. Its PDF shape depends critically on degrees of freedom (df): df = 1 has infinite density at 0; df = 2 is exponential; df ≥ 3 becomes unimodal with mode at df − 2 and variance 2df.

Prompt Grounding rubric:
level_1: No chi-square PDF curve ever appears on the white background, the df labels are absent or not integers, or fewer than two distinct df-labeled curves ever appear.
level_2: At least one curve appears with a legible df label, but not all five integer df values 1 through 5 ever get their own labeled curve, two curves are labeled with the same df value, or the plot's scale visibly changes between one df curve and the next.
level_3: Five separate curves appear, each carrying its own legible label reading df=1 through df=5 (one label per curve, all five integers present exactly once), all drawn on the same fixed white plotting area with a shared, unchanging horizontal and vertical scale throughout.
level_4: Additionally, the five curves appear or persist in ascending df order (1 before 2 before 3 before 4 before 5) with no label swapped or duplicated, every label stays attached to its own curve and legible for as long as that curve is on screen, and the frame contains only the five curves, their labels, and the white background.
level_5: Additionally, all five curves and their labels are simultaneously present and fully legible in the final frame, each curve's line style or color is distinguishable from every other so a viewer can tell them apart without relying on the labels alone, and no curve or label is cropped at the frame edge, tangled into illegibility, or missing from the final composite.

Scientific and Causal Correctness rubric:
level_1: None of the curves show the defining chi-square traits: e.g., all five look identical, the curves look symmetric/Gaussian-like rather than right-skewed, the shape doesn't appear to change with df at all, or the df=1 curve fails to rise sharply near x=0 (it looks like a rounded bump instead).
level_2: The curves are right-skewed and confined to x≥0 as expected, but key df-dependent features are wrong: e.g., the df=1 curve does not tower clearly above the others near x=0, the df=2 curve shows an interior hump instead of decreasing steadily from its highest point at x=0, or the df=3/4/5 curves don't show their peak shifting rightward as df increases.
level_3: The df=1 curve is tallest right at x=0 and falls off steeply; the df=2 curve is highest at x=0 and decreases the whole way with no interior bump; the df=3, df=4, and df=5 curves each show a rounded interior peak that shifts further right as df goes up (roughly around x=1, 2, and 3); no curve dips below zero or extends into negative x.
level_4: Additionally, near x=0 the df=1 curve sits clearly above the other four, while at larger x-values the higher-df curves (especially df=5) sit clearly above df=1: i.e., the curves' relative ranking visibly swaps as x increases rather than one curve staying on top everywhere; each curve is a single smooth, continuous line with no gaps, kinks, or jump discontinuities.
level_5: Additionally, as df rises from 1 to 5 each curve's overall spread visibly widens and its shape looks progressively less lopsided (less sharply right-skewed, more rounded/symmetric), matching how chi-square curves are known to behave as degrees of freedom grow, with no curve showing an extra bump, a flat plateau, or a shape that breaks this progressive trend.

Spatiotemporal Consistency rubric:
level_1: The white background changes color or vanishes partway through, a curve or its label suddenly jumps to a different on-screen position between one frame and the next, the plot's scale suddenly changes, a curve blinks fully on then off with no fade or transition, or the video freezes on a frame or cuts abruptly with no transition.
level_2: The background's brightness or tint shifts noticeably between one df curve appearing and the next, a curve's line thickness or color changes without reason, a curve's on-screen position drifts sideways or vertically by a noticeable amount (roughly more than 10% of the frame height) between frames, or a label momentarily disappears and comes back unchanged.
level_3: The background stays a constant, uniform white throughout; every curve, once drawn, keeps the same horizontal and vertical scale and position for the rest of the clip; no curve or label jumps location or drops out and returns; each new curve enters via a smooth fade, slide, or draw-on rather than popping in instantly.
level_4: Additionally, the spacing between curves and the overall layout stays proportionally constant as more curves are added, no curve appears stretched, skewed, or bent out of its plotted shape, motion of curves entering the scene looks continuous rather than jerky, and no shadow, reflection, or 3-D-style effect is applied to what should be a flat 2-D plot.
level_5: Additionally, once all five curves are present they hold their exact shapes, positions, and labels steady for the remainder of the clip with no visible flicker or shimmer along any curve's edge, and the pacing of the five curves' successive appearances feels even and continuous rather than speeding up, freezing, or skipping partway through.