Information hidden in raw data can be revealed most easily by means of statistical visualization techniques (“always plot your data”). To demonstrate I will now analyze the duration of reign (in years) for each of the 148 rulers of ancient Egypt and Nubia (“pharaohs”). Data are taken from https://www.metmuseum.org/toah/hd/phar/hd_phar.htm. High priests and monarchs who ruled less than one year are excluded. Now I could show you a table with 148 durations, but this would not afford much insight. Instead, we open the data set in JASP and create a simple histogram. When I did this, I somewhat expected to find the usual bell curve known as the Normal or Gaussian distribution. I was dead wrong:
[Note that in JASP the histogram can be constructed using several different procedures to determine the bin number and bin width type. Individual observations can be indicated by ‘rug’ tick marks, and a smooth estimate across the bins can be produced by selecting the option ‘Display density’.]
Firstly, these data are not bell-shaped at all. Instead, the distribution is asymmetric and resembles an exponential. In hindsight, this is not surprising; we are essentially looking at how long it takes for a “process” to fail, and failure times are generally modeled using exponential or Weibull distributions (for a more intricate modeling effort of the pharaoh data see Ramos et al., 2024; also note that Griffiths and Tenenbaum, 2006, propose an Erlang distribution for the pharaoh data). This underscores the generality of the field of statistics: the same statistical model can be applied to the reign of pharaohs, to accelerated life time tests for light bulbs, or to how many seasons it takes before Netflix pulls the plug on a series (see also Griffiths and Tenenbaum, 2006). Secondly, the data contain one obvious outlier: Pepi II, who reportedly reigned Egypt for no less than 94 years, from 2246-2152 BC. This is interesting also because the underlying cause for this outlier is unclear. On the one hand, Pepi II could be an exceptionally healthy (and lucky) individual; on the other hand, the outlier could simply be a recording/reporting error, perhaps due to a misreading. In fact, it has repeatedly been suggested that in actuality Pepi II reigned for “only” 64 years. Using JASP it is also easy to plot the duration of reigns as a time series, for consecutive pharaohs:
It is immediately apparent that pharao number 62 also reigned for an exceptionally long period: 66 years! This pharaoh was the great Ramses II (1279-1213 BC), who famously went into battle accompanied by his pet lion:
The lion battle of Ramses II. In a painting dated about 1907, Karl Oderich (1856-1915) shows the Pharaoh unleash his pet lion (named “Slayer of his Foes”) on the Hittite army at the battle of Kadesh in 1275 B.C.
A recent, advanced modeling effort for the pharaoh “duration of reign” data is presented in Ramos et al. (2024). I submit that the case of Pepi II is the earliest recorded “modern” outlier. There is some competition, as the Sumerian king list features entries such as Alulim of Eridu, who reigned for 28,800 years, and Alalngar, who reigned for 36,000 years. The differences with Pepi II are threefold: (a) there is solid evidence that Pepi II existed and ruled for at least 60 years; (b) the information on Pepi II is embedded in a long list of relatively reliable dates — a verifiable historical context; (c) the result for Pepi II is not biologically impossible. Nevertheless, for modeling the pharaoh duration of reign data, it seems prudent to follow modern scientific consensus and correct the reign of Pepi II from 94 years to 64 years.
References
Griffiths, T. L., & Tenenbaum, J. B. (2006). Optimal predictions in everyday cognition. Psychological Science, 17, 767-773.
Ramos, P. L., Jerez-Lillo, N., Segovia, F. A., Egbon, O. A., & Louzada, F. (2024). Power-law distribution in pieces: A semi-parametric approach with change point detection. Statistics and Computing, 34, 16.
Authors
Eric-Jan Wagenmakers (CEO) and the JASP Services BV Team. We are grateful to Michael Lee for a helpful comment on an earlier version of this post.
