SIM14 · 10 次左右选择 / 二项分布走向正态SIM14 · 10 LEFT/RIGHT CHOICES / BINOMIAL TO NORMAL
400 颗小球自己排出一条正态曲线Watch 400 balls arrange themselves into a normal curve
每颗球碰到一排钉，只做一次五五开的左或右选择。单次选择看不出形状；连续十次相加，再让 400 颗球重复同一过程，底部 11 格会逐步堆出钟形。动画不另掷随机，只重放种子 42 生成的 bit 序列。At each peg row, a ball makes one fifty-fifty left or right choice. A single choice has no shape; add ten choices and repeat the process for 400 balls, and the eleven bins build a bell curve. The animation never rerolls—it only replays the bit sequence generated from seed 42.
🎬 跟住一颗球：每碰一钉，按预先生成的 bit 左右一跳🎬 Track one ball: each peg replays one pre-generated left/right bit
SEED 42 · BIT 16
0 / 400
—
σ—
小球 / 观测频数Balls / observed count
二项理论曲线Binomial theory curve
逐球：同屏 6–10 颗Normal: 6–10 in flight快进：一次放完Fast: release all
💡 跑完看数据：11 格频数与理论中心对齐💡 Run it, then read the numbers: eleven observed bins center on theory
频数、均值、σ 均由 stats() 计算Counts, mean, and σ all come from stats()| 右跳次数 kRight jumps k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 观测频数Observed | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
观测均值 / 理论Observed mean / theory— / 5
观测 σ / 理论Observed σ / theory— / 1.581
已落球数Balls landed0
🎯 中心极限定理在现场🎯 The central limit effect on the board
很多个小的独立波动加在一起，整体才会走向正态；不是所有数据天生都该正态。Many small independent variations add toward normality. Normal is a mechanism, not a default label for every dataset.
⚠ 常见误区⚠ Common mistakes
✗ 数据都该是正态的✗ All data should be normal
✓ 只有“许多小独立因素相加”才给正态；一个大因素主导时直方图会偏斜，别硬套。✓ Normality emerges when many small independent causes add. One dominant cause can skew the histogram.
✗ 不正态 = 数据错了✗ Non-normal means bad data
✓ 先找那个大因素，而不是删数据。✓ Find the dominant cause before deleting observations.