Tag Archives: mathematics

Anscombe’s quartet

Anscombe’s quartet is four sets of data that are used to demonstrate the importance of graphing data.

Set 1 Set 2 Set 3 Set 4
x y x y x y x y
10 8.04 10 9.14 10 7.46 8 6.58
8 6.95 8 8.14 8 6.77 8 5.76
13 7.58 13 8.74 13 12.7 8 7.71
9 8.81 8 8.87 9 7.11 8 8.84
11 8.33 11 9.26 11 7.81 8 8.74
14 9.96 14 8.10 14 8.84 8 7.04
6 7.24 6 6.13 6 6.08 8 5.25
4 4.26 4 3.10 4 5.39 19 12.5
12 10.8 12 9.13 12 8.15 8 5.56
7 4.82 7 7.26 7 6.42 8 7.91
5 5.68 5 4.74 5 5.73 8 6.89
Mean 9 7.50 9 7.50 9 7.50 9 7.50
Variance 11 4.13 11 4.13 11 4.12 11 4.12
PMCC 0.82 0.82 0.82 0.82

Each set of data has near-identical statistical properties: the same average and variance (for both x and y), and the same product moment correlation coefficient and linear regression line. When plotted, however, they look entirely different. (The scale of the last graph is different from the others.)

You can download Anscombe’s quartet as an Excel spreadsheet.

Francis Anscombe, “Graphs in Statistical Analysis”, American Statistician 27(1) (1973): 17‑21. http://www.jstor.org/stable/2682899 (.PDF).

Haversine formula

The haversine formula is used to calculate the distance between two points on the Earth’s surface specified in longitude and latitude.

d is the distance between two points with longitude and latitude (ψ,φ) and r is the radius of the Earth.

As an example I have calculated the distance between Fermilab in Illinois (41° 49′ 55″ N, 88° 15′ 26″ W) and CERN’s Meyrin campus in Switzerland (46° 14′ 3″ N, 6° 3′ 10″ E). There’s a little too much maths for this site to handle so I have included a .PDF file of the working below.

The value calculated is 7084 km, which isn’t quite correct. This is because the formula assumes that the Earth is a perfect sphere when in fact it is an oblate spheroid. To compensate for this Vincenty’s Formulae must be used; these are much more complicated but give a more accurate value of 7103 km.

Spherical ice cubes and surface area to volume ratio

I’ve recently been experimenting with making spherical ice cubes for cocktails.

But why go to all the fuss of making spherical ice cubes? What’s wrong with regular ice cubes? The answer is surface area to volume ratio: the volume of the ice provides the cooling effect but the surface area controls how fast the ice melts – the lower the surface area to volume ratio the longer the ice will take to melt for the same cooling effect. Essentially, a lower surface area to volume ratio keeps your drink cold, but stops it from becoming too diluted.

A cube with sides of length x will have a volume of x3 and a surface area of 6x2. The surface area to volume ratio for a cube is therefore 6 to 1 (6:1). Of all the Platonic solids (solids with identical faces) the icosahedron has the lowest surface area to volume ratio.

Of all the regular shapes a sphere has the lowest possible surface area to volume ratio. That is what makes it particularly well suited for cooling drinks.

The production of spherical ice cubes is also quite interesting. They’re usually made in an extremely elaborate process using large blocks of ice that are then shaped using metal “presses” (usually made of copper or aluminium as they are very good conductors of heat).

Percent, permil and basis points

I only recently discovered the permil (cf. percent), a typographic character that enables you to give a fraction equal to one part in one thousand without using a decimal point. For example 12.3% = 123‰ (“twelve-point-three percent is equal to one hundred and twenty-three permil”).

There is also a symbol (‱) for basis points (aka permyriad), parts in ten thousand. For example 12.34% = 123.4‰ = 1234‱ (“twelve-point-three-four percent is equal to one hundred and twenty-three-point-four permil or one thousand, two hundred and thirty-four basis points”).

A large number of fonts are unable to render the permil and/or basis point symbols correctly, so the post above may be missing some symbols.

How far away is the horizon?

It’s relatively easy to calculate how far away the horizon is if you know two things: the height of your eye above the ground, and how big the Earth is.

Because we know r, the radius of the Earth, and we can measure h, the height of the eye above the ground, we can use Pythagorus’s theorem to calculate d, the distance to the horizon.

Which gives an equation for the distance to the horizon d as a function of r and h:

For the average person’s height of 1.62 m and the average radius of the Earth of 6367.5 km that gives a distance to the horizon of 4542 metres or 2.8 miles.

From the world’s highest public observation deck, on the 100th floor of the Shanghai World Financial Centre at a height of 474 metres the distance to the horizon is 77.8 kilometres, giving a viewable area of nineteen billion square metres, over seven thousand square miles.