Domain: Natural Science — Statistics & Data Science
Target concept: Plot the binomial distribution graphs for n=10 and n=100, and verify that the graph for n=100 is more bell-shaped and symmetric, whereas the graph for n=10 is discrete.

Video generation prompt: Demonstrate a simple statistical example: generate two binomial distributions with parameter p=0.5 on a computer screen. First, set n=10 and plot the histogram; then set n=100 and plot the histogram. Display the two charts side by side clearly for comparison, using the same y-axis scale (probability) for both while letting each histogram's x-axis span its own natural range, and employing standard plotting software procedures.

Evaluation guide:
(i) Target concept: Visual demonstration of convergence of Binomial(n,p=0.5) to Normal via increasing n.
Minimum mechanism: Correct computation and rendering of binomial probability mass function for integer k ∈ [0,n]; identical normalization and y-axis (probability) scaling across plots, with each plot's x-axis spanning its own natural range (0–10 for n=10, 0–100 for n=100); side-by-side spatial registration enabling direct shape comparison.
(ii)Storyline: Frame 1–3: Computer screen appears with blank plotting environment. Frame 4–6: Left histogram (n=10) renders fully: 11 distinct bars, labeled “n=10”. Frame 7–9: Right histogram (n=100) renders fully: dense, bell-shaped bar cluster, labeled “n=100”. Frame 10: Stable final frame showing both plots simultaneously, aligned side by side, each on its own natural x-range with a shared probability scale, with no occlusion or animation: fixed viewpoint, front-facing, full-chart visibility.

Manual:
Experimental Procedure
1. Launch plotting software (e.g., Python matplotlib or R ggplot2) on a clean computer screen.
2. Generate binomial PMF histogram for n=10, p=0.5; display as left plot.
3. Generate binomial PMF histogram for n=100, p=0.5; display as right plot.
4. Arrange both plots side-by-side, each using its own natural x-axis range (0–10 for n=10, 0–100 for n=100) but sharing the same y-axis quantity and scaling (probability).
Observation of Experimental Phenomena
- Left plot shows 11 discrete, separated bars spanning x=0 to 10, symmetric about x=5.
- Right plot shows ~101 densely packed bars spanning x=0 to 100, forming a smooth, symmetric, bell-shaped curve centered at x=50.
- Both plots share identical y-axis scaling, labeling convention, and visual styling, even though their x-axis ranges differ to match each plot's own support.
Summary of Experimental Principles
Binomial distribution with p=0.5 becomes increasingly symmetric and bell-shaped as n increases due to the Central Limit Theorem; n=10 remains visibly discrete while n=100 approximates a normal distribution.

Prompt Grounding rubric:
level_1: No binomial histograms appear at all; there is no side-by-side layout, no 'n=10' or 'n=100' label anywhere, and no sense that this is being shown on a computer screen or plotting environment.
level_2: At least one histogram is visible, but it is not labeled n=10 or n=100, the two histograms are not placed next to each other, or one of the two charts is simply missing.
level_3: Two histograms are shown side by side on a screen, correctly labeled 'n=10' and 'n=100'; both have visible x- and y-axes, but something about their scaling breaks fair comparison: for example the y-axis shows counts on one chart and probability on the other, or the tick spacing within one chart looks arbitrary and uneven (note: the n=10 chart naturally spanning roughly 0–10 while the n=100 chart spans roughly 0–100 is expected and not itself a flaw).
level_4: All conditions for score 3 are met; in addition, the n=10 histogram's x-axis spans roughly 0 to 10 and the n=100 histogram's x-axis spans roughly 0 to 100 (each plot's own natural range), both y-axes represent the same quantity (e.g., both show probability, not one count and one probability) so bar heights are visually comparable in meaning, and every bar, axis, and label in both charts is fully visible with no UI element covering either chart.
level_5: All conditions for score 4 are met; in addition, the n=10 plot clearly shows a small number of separated, blocky bars while the n=100 plot shows a dense cluster of many thin bars forming a smooth-looking outline; both plots share the same bar color, label font, background, and overall chart style; and there are no rendering glitches such as clipped bars or misaligned tick marks in either chart.

Scientific and Causal Correctness rubric:
level_1: The two charts don't look like binomial distributions at all, e.g., one looks uniform (flat bars), strictly increasing/decreasing, or skewed in a way that doesn't match p=0.5; or the two charts show no visible difference in shape or spread from each other.
level_2: Both charts resemble histograms, but at least one violates basic binomial-with-p=0.5 shape: the n=10 chart's bars extend past x=0..10 or clearly aren't symmetric around x=5, or the n=100 chart's bars are lumpy/bimodal rather than bell-shaped; there is little visible contrast between a blocky n=10 shape and a smooth n=100 shape.
level_3: The n=10 chart shows a small number of bars roughly symmetric around its center (x=5), tallest near the middle and shorter toward the edges; the n=100 chart shows many bars forming a bell-shaped outline roughly symmetric around its center (x=50); however, the y-axis meaning is inconsistent between the two charts (e.g., one plots raw counts and the other probability) so they cannot be directly compared.
level_4: All conditions for score 3 are met; in addition, the n=10 chart shows bars for every integer from 0 to 10 with heights decreasing smoothly moving away from x=5 in both directions; the n=100 chart shows bars densely covering the range where probability is visually non-negligible, forming a smooth, symmetric, bell-shaped envelope; both charts use the same y-axis quantity (probability) so their shapes are directly comparable.
level_5: All conditions for score 4 are met; in addition, the relative heights within each chart visibly follow the expected binomial pattern: for the n=10 chart, the tallest bars are clearly at x=5 with a smooth symmetric taper on both sides; for the n=100 chart, the bell shape looks visibly narrower relative to its own axis range than the n=10 chart's spread is relative to its axis range, making the sharper, more concentrated bell of the larger n unmistakable; the blocky look of the n=10 chart versus the smooth look of the n=100 chart is unambiguous to a viewer.

Spatiotemporal Consistency rubric:
level_1: One or both charts appear or disappear abruptly with no transition; a chart flickers or jumps to a different position; axis labels jump from one location to another between frames; the side-by-side layout breaks apart mid-clip; or there is a visible tear or split in the image.
level_2: The side-by-side layout is established, but one chart resizes or shifts position during the clip; tick labels move between frames; a chart's border warps or bends; or the background changes unexpectedly partway through.
level_3: Both charts stay fixed in position and size for the whole clip; no bars, axes, or labels move on their own; there is no warping or stretching of either chart; however, there is a minor, easy-to-miss flaw such as brief jitter in the text of one label.
level_4: All conditions for score 3 are met; in addition, neither chart translates, rotates, rescales, or deforms at any point; all text stays sharp and stationary; there is no flicker or blinking of any bar or label; and the axis lines and bar edges of both charts stay consistently aligned frame to frame.
level_5: All conditions for score 4 are met; in addition, both charts stay in exactly the same screen position, size, and framing for the entire clip with no visible drift, shimmer, or blur on either chart; there is no change in zoom, crop, or viewing angle; and the overall image looks consistently sharp and stable from the first frame to the last.