# Cavendish's Determinations of the Density of the Earth

### Description

Henry Cavendish carried out a series of experiments in 1798 to determine the
mean density of the earth, as an indirect means to calculate the
gravitational constant, G, in Newton's formula for the force (f) of
gravitational attraction,
*f = G m M / r^2*
between two bodies of mass m and M.

Stigler (1977) used these data to illustrate properties of robust estimators with real, historical data. For these data sets, he found that trimmed means performed as well or better than more elaborate robust estimators.

### Usage

1 |

### Format

A data frame with 29 observations on the following 3 variables.

`density`

Cavendish's 29 determinations of the mean density of the earth

`density2`

same as

`density`

, with the third value (4.88) replaced by 5.88`density3`

same as

`density`

, omitting the the first 6 observations

### Details

Density values (D) of the earth are given as relative to that of water. If the earth is
regarded as a sphere of radius R, Newton's law can be expressed as
*G D = 3 g / (4 π R)*, where *g=9.806 m/s^2* is the acceleration due to gravity; so G is proportional to 1/D.

`density`

contains Cavendish's measurements as analyzed, where he treated the
value 4.88 as if it were 5.88. `density2`

corrects this.
Cavendish also changed his experimental apparatus after the sixth determination,
using a stiffer wire in the torsion balance. `density3`

replaces the first
6 values with `NA`

.

The modern "true" value of D is taken as 5.517.
The gravitational constant can be expressed as *G = 6.674 * 10^-11 m^3/kg/s^2*.

### Source

Kyle Siegrist, "Virtual Laboratories in Probability and Statistics", http://www.math.uah.edu/stat/data/Cavendish.html

Stephen M. Stigler (1977), "Do robust estimators work with *real* data?", *Annals of Statistics*, 5, 1055-1098

### References

Cavendish, H. (1798). Experiments to determine the density of the earth.
*Philosophical Transactions of the Royal Society of London*, 88 (Part II),
469-527.
Reprinted in A. S. Mackenzie (ed.), *The Laws of Gravitation*, 1900,
New York: American.

Brownlee, K. A. (1965). *Statistical theory and methodology in science and engineering*,
NY: Wiley, p. 520.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
data(Cavendish)
summary(Cavendish)
boxplot(Cavendish, ylab='Density', xlab='Data set')
abline(h=5.517, col="red", lwd=2)
# trimmed means
sapply(Cavendish, mean, trim=.1, na.rm=TRUE)
# express in terms of G
G <- function(D, g=9.806, R=6371) 3*g / (4 * pi * R * D)
boxplot(10^5 * G(Cavendish), ylab='~ Gravitational constant (G)', xlab='Data set')
abline(h=10^5 * G(5.517), col="red", lwd=2)
``` |