# History: the role of statistics in astronomy

I recently boldly claimed in front of a group of fairly smart eighth grade students that astronomy contributed greatly to the foundations of statistics and many statistical concepts were invented for use in astronomy. However, looking to back that up, I was fairly disappointed. Errors, the mean and the median deviation from the mean may have been first observed in astronomy. However, even the concept of error propagation might stem more from classical mechanics than astronomy. Beyond these concepts, I was unable to find much more. Feigelson writes (http://arxiv.org/pdf/astro-ph/0401404.pdf):

Ptolemy estimated parameters of a non-linear cosmological model using a minimax goodness-of-fit method. Al-Biruni discussed the dangers of propagating errors from inaccurate instruments and inattentive observers. While some Medieval scholars advised against the acquisition of repeated measurements, fearing that errors would compound rather than compensate for each other, the usefulnes of the mean to increase precision was demonstrated with great success by Tycho Brahe.

Can you suggest good references that have some more details on the historical links between astronomy and statistics?

Thank you for the excellent answers!

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I recommend to you the book "The Lady Tasting Tea" for the sources of many of the statistical methods currently used. They find a majority of their original sources in agriculture - do deal with extensive noise. I am not familiar with astronomical phenomena being characterized as being so full of noise that statistical methods are required to bring analytical form and order to them. –  EngrStudent Jul 3 '14 at 2:52
Sorry to dissent, but that book is deeply unreliable. I gave many details in a review in Biometrics 57: 1273-1274 (2001). Much better sources are the books by Anders Hald and Stephen Stigler. –  Nick Cox Jul 3 '14 at 7:29
Laplace used inverse (i.e. Bayesian) probability to give margins of error on the mass of Saturn. The Le Verrier/Adams projections which led to the discovery of Neptune were effective a form of regression. –  Henry Jul 3 '14 at 11:32

The main source is Stephen M. Stigler, The History of Statistics, Part One, "The Development of Mathematical Statistics in Astronomy and Geodesy before 1827". Another useful source is John Aldrich, Figures from the History of Probability and Statistics.

You could also look at Searle, Casella and McCulloch, Variance Components, chap. 2:

• p. 23: The method of least squares was independently discovered by Legendre and Gauss. The story is told by R.L. Plackett, "Studies in the History of Probability and Statistics. XXIX: The Discovery of the Method of Least Squares", Biometrika, 59, 239-251.

• p. 24: According to R.D. Anderson, "astronomers understood the concept of degrees of freedom (but without using the term) as early as the year 1852". He refers to B. J. Peirce, "Criterion for the rejection of doubtful observations", The Astronomical Journal, 2, 161-163 (see here), who specified "the sum of squares of all errors' as being $(N-m)\varepsilon^2$, where $N$ is the total number of observations, $m$ is the number of unknown quantities contained in the observations and $\varepsilon^2$ is the mean error (sample variance)."

• pages 23-24: The first formulation of a random effects model is that of George Biddell Airy, in a monograph published in 1861. See also Marc Nerlove, "The History of Panel Data Econometrics, 1861-1997", in Essays in Panel Data Econometrics: "what Airy calls a Constant error, we would call a random day effect". It is the error that remains even when every known instrumental correction has been applied.

• pages 24-25: The second use of a random effects model appears in W. Chauvenet, A Manual of Spherical and Practical Astronomy, 2: Theory and Use of Astronomical Instruments, 1863. He derived the variance of $\bar{y}_{..}=\sum_{i=1}^a\sum_{j=1}^n y_{ij}/an$ as $$\text{var}(\bar{y}_{..})=\frac{\sigma^2_a+\sigma^2_e/n}{a}$$

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Probably the best-known example of a statistical method "developed" from an astronomy problem was Gauss' use of least squares to generate an orbit for Ceres on the basis of Piazzi's observations. Piazzi did not have nearly enough observations for conventional methods of determining orbits when Ceres was lost in the glare of the sun. Gauss took the data, applied least squares and told the astronomers where to point their telescopes to find it again. See Forbes, 1971 "Gauss and the discovery of Ceres", J of the History of Astronomy.

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