Domain: Natural Science — Statistics & Data Science
Target concept: Standard normal distribution (μ = 0, σ = 1)

Video generation prompt: A white background displays the probability density function of a normal distribution with (μ=0, σ=1). Then, keeping the same scale, σ is incrementally increased by 1 up to 5, and the changes in the probability density function are observed.

Evaluation guide:
(i) Target concept: Visual representation of how standard deviation σ governs the horizontal spread and vertical height of a normal probability density function under fixed mean μ=0.
Minimum mechanism: Curves must be mathematically derived from f(x) = (1/(σ√(2π)))·exp(−x²/(2σ²)), ensuring inverse proportionality between σ and peak height, and direct proportionality between σ and full width at half maximum.
(ii)Storyline: Frame 0–1: White background appears. Frame 2–3: σ=1 curve fades in, centered, narrow and tall. Frame 4–5: σ=2 curve appears (either beside or replacing), visibly wider and shorter. Frame 6–7: σ=3, then σ=4, then σ=5 follow in strict ascending order, each wider and shorter than prior, all sharing x=0 center. Final frame: All five curves coexist or final σ=5 dominates, with unambiguous σ labels, fixed viewpoint, and no visual contradiction to monotonic σ-scaling. Perspective: Frontal, static, orthographic (no perspective distortion); visibility: All curves and labels fully in-frame, no occlusion, no glare, no transparency effects.

Manual:
Experimental Procedure
1. Display white background.
2. Render probability density curve for standard normal distribution (μ=0, σ=1).
3. Sequentially or simultaneously render four additional curves, each with σ = 2, 3, 4, 5 — maintaining same vertical/horizontal scale and μ=0 centering.
Observation of Experimental Phenomena
- All curves are symmetric, bell-shaped, and centered at x=0.
- As σ increases from 1 to 5, each curve becomes progressively wider and shorter.
- Curve tails extend farther horizontally with higher σ.
Summary of Experimental Principles
- The standard normal distribution has μ=0, σ=1.
- Scaling σ controls spread: larger σ → greater variance → flatter, wider PDF.

Prompt Grounding rubric:
level_1: No bell-shaped probability curve is ever shown, the white background is missing, or none of the σ values 1 through 5 appear as labels anywhere in the video.
level_2: The white background is present with at least one bell-shaped curve, but fewer than five distinct curves ever appear, no numeric σ label ties any curve to a specific σ value, or a shared center reference for μ=0 is never indicated (the curves' peaks don't line up over a common x-position).
level_3: Five bell-shaped curves appear (together or one after another) against the white background, each carrying a legible σ label spanning 1 through 5, and every curve's peak sits above the same x-position, implying the shared μ=0.
level_4: Additionally, all five σ labels (1, 2, 3, 4, 5) are simultaneously legible and unambiguously paired to their own curve, with no label swapped, duplicated, or missing, and no stray text, axis, or gridline beyond what's needed to display the five curves.
level_5: Additionally, the frame composition stays fixed for the entire clip (no panning, zooming, or rotating of the plotting area), and all five curves and their labels stay fully on-screen for as long as they're meant to be visible, with no cropping, no label hidden behind another curve, and no gap in the σ=1 through σ=5 progression.

Scientific and Causal Correctness rubric:
level_1: The curve shapes are not bell-like Gaussian forms (e.g., triangular, uniform, flat, or lopsided), the curves visibly narrow as the labeled σ value increases (the reverse of the true relationship), or the curves' peaks are not aligned over the same x-position despite the shared μ=0 setup.
level_2: The curves are roughly bell-shaped, but the σ-to-width/height relationship is inconsistent: for example the σ=3 curve isn't visibly wider or shorter than the σ=2 curve, or two differently labeled σ curves look identical in width and height.
level_3: Each curve is a smooth, symmetric bell shape, and moving from σ=1 to σ=5 the curve visibly widens and its peak visibly drops at every step (σ=1 is the narrowest and tallest, σ=5 the widest and shortest), with no step reversing this trend.
level_4: Additionally, each curve's tails visibly stretch farther from center as σ increases (the σ=5 curve reaches noticeably farther out before flattening to near-zero than the σ=1 curve does), and no curve shows a shape inconsistent with a Gaussian, such as a flat top, a secondary bump, or a sharp corner.
level_5: Additionally, the width-and-height change reads as gradual and continuous across all five steps (each increase from σ=k to σ=k+1 produces a comparable, proportionate reduction in peak height and increase in spread rather than one uneven jump), and the σ=1 curve stays recognizably the tallest/narrowest while σ=5 stays recognizably the shortest/widest whenever multiple curves are shown together.

Spatiotemporal Consistency rubric:
level_1: Curves appear or vanish abruptly with no transition, the plotting area jumps position or scale between σ steps, or the background changes color or brightness mid-sequence.
level_2: A curve flashes in or out rather than fading or sliding in smoothly, the background brightness noticeably shifts between σ steps, or a curve's on-screen position shifts sideways or vertically between frames without any transition motion.
level_3: The white background and the plot's overall position stay constant throughout; each curve that enters does so via a smooth fade or slide rather than a sudden pop; no curve or label ever disappears partway through and comes back later; the viewpoint stays static and frontal for the whole clip.
level_4: Additionally, curves that overlap in the final frame remain fully and simultaneously visible without one curve intermittently blinking out from behind another, entrance timing looks consistent across the five curves rather than some appearing much faster than others, and no curve visibly shears, stretches, or distorts while appearing.
level_5: Additionally, once a curve settles into place it holds a steady shape and position for the remainder of the clip with no drifting, wobbling, or flickering, the σ labels stay attached to their correct curve without sliding away, and the overall σ=1→5 progression reads as one continuous, evenly-paced sequence with no stutter or repeated/skipped frame.