# Central Limits (DD-588)

> The Galton board, also called the quincunx or bean machine, has interleaved rows of pegs above a row of bins. Each bead that falls through bounces left or right at each peg. The number of paths to each bin follows Pascal's triangle, so the bead counts follow a binomial distribution. With many rows and many beads, that binomial distribution approaches the normal distribution, the bell curve. That's a special case of the central limit theorem known as the de Moivre–Laplace theorem. Francis Galton designed the board for a Royal Institution discourse on February 27, 1874. He described a two-stage version in an 1877 letter to George Darwin and celebrated the 'Law of Frequency of Error' in his 1889 book Natural Inheritance. He used it to illustrate his ideas about heredity and regression toward the mean ('reversion to mediocrity'), ideas that were part of his eugenics. The bell shape isn't automatic. A tilted board or biased pegs shift and skew it, and other peg shapes give other distributions: Jacobus Kapteyn built a board that produces a log-normal curve. Large Galton boards designed by Charles and Ray Eames appeared in the IBM Pavilion at the 1964–65 New York World's Fair. The footage shows a modern desktop Galton board with a printed bell curve, its beads settling into the shape.

- Department: Science
- First published: 2025-06-03
- Last updated: 2025-06-03
- Permalink: https://detailenjoyer.com/case/central-limits/
- Author: Marcus Easterling (DetailEnjoyer), The Detail Department

## On the timeline

- **Feb. 27, 1874**: Galton presents his board at a Royal Institution discourse
- **1877**: Galton describes a two-stage board in a letter to George Darwin
- **1964**: Eames-designed Galton boards appear at the IBM Pavilion at the New York World's Fair

Full timeline: https://detailenjoyer.com/timeline/?case=central-limits

## Quick facts

- **DD-588-F01**: The number of paths to each bin of a Galton board follows Pascal's triangle. _(Write-up · Exhibit A (Left or right))_
- **DD-588-F02**: Francis Galton designed his board for a Royal Institution discourse in 1874. _(Write-up · Exhibit C (Galton's 1874 lecture))_

## Write-up

Random bounces, one bell curve. Francis Galton showed his board at the Royal Institution in 1874.

1. **Left or right**: Each bead bounces left or right at every peg, and the bin counts follow a binomial distribution.
2. **Toward the bell**: With many rows, the binomial approaches the normal distribution (the de Moivre–Laplace theorem).
3. **Galton's 1874 lecture**: Galton designed the board for a Royal Institution discourse on February 27, 1874.
4. **Heredity and eugenics**: Galton used it to illustrate regression toward the mean, part of his eugenic ideas.
5. **Not always a bell**: Biased pegs or other peg shapes give other distributions, such as Kapteyn's log-normal board.

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## Reel vs. record

- **Note**: 'Demonstrate the central limit theorem': fair, with one detail. Strictly, it shows the de Moivre–Laplace theorem, the special case where a binomial distribution approaches the normal.
- **Note**: 'Regardless of the underlying random process at each peg': I'd call that an overstatement. The bell needs many roughly independent bounces. A tilted board or biased pegs shift and skew it, and differently shaped pegs can produce other curves, such as Kapteyn's log-normal board.
- **Missing**: Galton designed the board for a lecture on February 27, 1874, and used it to argue about heredity and regression toward the mean, ideas tied to his eugenics.

## Video: Central Limits: Random Bounces, One Bell Curve

- Duration: 23.3 s · Published: 2025-06-03
- File: https://detailenjoyer.com/assets/videos/DKdOvMTSc2o.mp4

### Transcript

DID YOU KNOW? Galton's Board is a device invented by Sir Francis Galton to visually demonstrate the central limit theorem, and the emergence of the normal distribution. It consists of a vertical board with rows of evenly spaced pegs; as balls are dropped from the top, they bounce randomly left or right as they hit each peg. After passing through multiple rows, the balls accumulate in bins at the bottom, forming a bell- shaped curve, regardless of the underlying random process at each peg. NOW YOU KNOW!

## Questions & answers

**Q: What does a bead do at each peg of a Galton board?**  
A: The counts follow a binomial distribution. _(Source: Write-up · Exhibit A (Left or right) · Wikipedia: Galton board; Four Pines Publishing: The Galton Board)_

**Q: What shape do the beads form with many rows and beads?**  
A: The normal distribution. _(Source: Write-up · Exhibit B (Toward the bell) · Wikipedia: Galton board; Kunert, Montag & Pöhlmann, The quincunx: history and mathematics, Statistical Papers (2001))_

**Q: When did Galton show his board at the Royal Institution?**  
A: On February 27, 1874. _(Source: Write-up · Exhibit C (Galton's 1874 lecture) · Wikipedia: Galton board; Kunert, Montag & Pöhlmann, The quincunx: history and mathematics, Statistical Papers (2001))_

**Q: What idea did Galton use the board to illustrate?**  
A: He called it 'reversion to mediocrity'. _(Source: Write-up · Exhibit D (Heredity and eugenics) · Wikipedia: Galton board; Kunert, Montag & Pöhlmann, The quincunx: history and mathematics, Statistical Papers (2001))_

**Q: What did Jacobus Kapteyn's board produce?**  
A: Its pegs 'multiply' instead of 'sum'. _(Source: Write-up · Exhibit E (Not always a bell) · Wikipedia: Galton board)_

## Sources

- [Wikipedia: Galton board](https://en.wikipedia.org/wiki/Galton_board)
- [Kunert, Montag & Pöhlmann, The quincunx: history and mathematics, Statistical Papers (2001)](https://doi.org/10.1007/s003620100048)
- [Four Pines Publishing: The Galton Board](https://www.galtonboard.com/)

## Cite this case file

Easterling, Marcus. "Central Limits." The Detail Department (DetailEnjoyer), case file DD-588, last updated Jun 3, 2025. https://detailenjoyer.com/case/central-limits/