# quincunx2()

# Demonstration of the Galton Box, example 2

### Yihui Xie & Lijia Yu / 2017-04-04

Demonstration of the Quincunx (Bean Machine/Galton Box)

Simulates the quincunx with ‘balls’ (beans) falling through several layers (denoted by triangles) and the distribution of the final locations at which the balls hit is denoted by a histogram.

`quincunx()`

is used to model intergenerational variation: balls are dropped from
the top and cascade randomly through rows of alternating offset pins, landing in
compartments at the bottom as a binomial or approximately normal distribution^{1}.

`quincunx2()`

is afforded insight into regression to the mean. When the pellets
in an upper compartment are released, their average final position is directly
below. But what if we ask of a compartment at the lower level, from where did
these pellets come? The answer was not ‘on average, directly above’. Rather,
it was ‘on average, more towards the middle’, for the simple reason that there
were more pellets above it towards the middle that could wander left than there
were in the left extreme that could wander to the right, inwards^{1}.

The bean machine, also known as the quincunx or Galton box, is a device invented by Sir Francis Galton to demonstrate the law of error and the normal distribution.

When a ball falls through a layer, it can either go to the right or left side with the probability 0.5. At last the location of all the balls will show us the bell-shaped distribution.

```
library(animation)
set.seed(123)
ani.options(nmax = 200 + 15 - 2, 2)
freq = quincunx2(balls = 200, col.balls = rainbow(200))
```

```
## frequency table
barplot(freq$top, space = 0) # top layers
```

```
barplot(freq$bottom, space = 0) # bottom layers
```

- Darwin, Galton and the Statistical Enlightenment, http://onlinelibrary.wiley.com/doi/10.1111/j.1467-985X.2010.00643.x/full ↩