Statistical Science 2014, Vol. 29, No. 1, 106–112 DOI: 10.1214/13-STS417 c Institute of Mathematical Statistics, 2014 The Two-Piece Normal, Binormal, or Double Gaussian Distribution: Its Origin and Rediscoveries Kenneth F. Wallis Abstract. This paper traces the history of the two-piece normal distri- bution from its origin in the posthumous Kollektivmasslehre (1897) of Gustav Theodor Fechner to its rediscoveries and generalisations. The denial of Fechner’s originality by Karl Pearson, reiterated a century later by Oscar Sheynin, is shown to be without foundation. Key words and phrases: Gustav Theodor Fechner, Gottlob Friedrich Lipps, Francis Ysidro Edgeworth, Karl Pearson, Francis Galton, Oscar Sheynin. 1. INTRODUCTION tics (Johnson and Kotz (1970)), in which the two- piece normal distribution made no appearance, un- The two-piece normal distribution came to pub- der this or any other name. On the contrary, the lic attention in the late 1990s, when the Bank of distribution was originally introduced in Fechner’s England and the Sveriges Riksbank began to pub- Kollektivmasslehre (1897) as the Zweispaltiges or lish probability forecasts of future inflation, using Zweiseitige Gauss’sche Gesetz. In his monumental this distribution to represent the possibility that the history of statistics, Hald (1998) prefers the latter balance of risks around the central forecast might name, which translates as the “two-sided Gaussian not be symmetric. The forecast probabilities that law,” and refers to it as “the Fechner distribution” future inflation would fall in given intervals could (page 378). However Fechner’s claim to originality be conveniently calculated by scaling standard nor- had been disputed by Pearson (1905), whose denial mal probabilities, and the resulting density forecasts of Fechner’s originality has recently been repeated were visualised in the famous forecast fan charts. by Sheynin (2004). In this paper we reappraise the In both cases the authors of the supporting tech- source and nature of the various claims, and record arXiv:1405.4995v1 [stat.ME] 20 May 2014 nical documentation (Britton, Fisher and Whitley, several rediscoveries of the distribution and exten- 1998; Blix and Sellin (1998)) refer readers to John- sions of Fechner’s basic ideas. As a prelude to the son, Kotz and Balakrishnan (1994) for discussion of discussion, there follows a brief technical introduc- the distribution. These last authors state (page 173) tion to the distribution. that “the distribution was originally introduced by A random variable X has a two-piece normal Gibbons and Mylroie (1973),” a reference that post- distribution with parameters µ,σ1 and σ2 if it has dates the first edition of Distributions in Statis- probability density function (PDF) 2 2 Kenneth F. Wallis is Emeritus Professor of A exp[ (x µ) /2σ1], x µ, Econometrics, Department of Economics, University of (1) f(x)= − − 2 2 ≤ A exp[ (x µ) /2σ2], x µ, Warwick, Coventry CV4 7AL, United Kingdom e-mail: − − ≥ −1 [email protected]. where A = (√2π(σ1 + σ2)/2) . The distribution is This is an electronic reprint of the original article formed by taking the left half of a normal distribu- published by the Institute of Mathematical Statistics in tion with parameters (µ,σ1) and the right half of Statistical Science, 2014, Vol. 29, No. 1, 106–112. This a normal distribution with parameters (µ,σ2), and reprint differs from the original in pagination and scaling them to give the common value f(µ)= A at typographic detail. the mode, µ, as in (1). The scaling factor applied to 1 2 K. F. WALLIS −1 −1 quantiles xp = F (p) and zp =Φ (p). Then in the left piece of the distribution we have xα = σ1zβ + µ, where β = α(σ1 + σ2)/2σ1. And in the right piece of the distribution, defining quantiles with reference to their upper tail probabilities, we have x1−α = σ2z1−δ + µ, where δ = α(σ1 + σ2)/2σ2. In particu- lar, with σ1 <σ2, as in Figure 1, the median of the −1 distribution is x0.5 = σ2Φ (1 (σ1 + σ2)/4σ2)+ µ. In this case the three central− values are ordered mean>median>mode; with negative skewness this order is reversed. Although the two-piece normal PDF is continuous at µ, its first derivative is not and the second deriva- Fig. 1. The probability density function of the two-piece nor- tive has a break at µ, as first noted by Ranke and mal distribution. Dashed line: left half of N(µ, σ1) and right Greiner (1904). This has the disadvantage of making half of N(µ, σ2) distributions with µ = 2.5 and σ1 <σ2. Solid line: the two-piece normal distribution. standard asymptotic likelihood theory inapplicable, nevertheless standard asymptotic results are avail- able by direct proof for the specific example. the left half of the N(µ,σ1) PDF is 2σ1/(σ1 + σ2) The remainder of this paper is organised as fol- while that applied to the right half of N(µ,σ2) is lows. In Section 2 we revisit the distribution’s origin 2σ2/(σ1 + σ2), so the probability mass under the in Gustav Theodor Fechner’s Kollektivmasslehre, left or right piece is σ1/(σ1 + σ2) or σ2/(σ1 + σ2), edited by Gottlob Friedrich Lipps and published in respectively. An example with σ1 <σ2, in which the 1897, ten years after Fechner’s death. In Section 3 we two-piece normal distribution is positively skewed, note an early rediscovery, two years later, by Fran- is shown in Figure 1. The skewness becomes extreme cis Ysidro Edgeworth. In Section 4 we turn to the as σ1 0 and the distribution collapses to the half- normal→ distribution, while the skewness is reduced first discussion in the English language of Fechner’s contribution, in a characteristically long and argu- as σ1 σ2, reaching zero when σ1 = σ2 and the dis- tribution→ is again the normal distribution. mentative article by Karl Pearson (1905). Pearson The mean and variance of the distribution are derives some properties of “Fechner’s double Gaus- sian curve,” but asserts that it is “historically in- 2 (2) E(X)= µ + (σ2 σ1), correct to attribute [it] to Fechner.” We re-examine rπ − Pearson’s evidence in support of this position, in 2 2 particular having in mind its reappearance in Os- (3) var(X)= 1 (σ2 σ1) + σ1σ2. − π − car Sheynin’s (2004) appraisal of Fechner’s statis- tical work. Pearson also argues that “the curve is Expressions for the third and fourth moments about not general enough,” especially in comparison with the mean are increasingly complicated and unin- his family of curves. The overall result was that the formative. Skewness is more readily interpreted in Fechner distribution was overlooked for some time, terms of the ratio of the areas under the two pieces to the extent that there have been several indepen- of the PDF, which is σ1/σ2, or a monotone transfor- dent rediscoveries of the distribution in more recent mation thereof such as (σ2 σ1)/(σ2 + σ1), which is years; these are noted in Section 5, together with the value taken by the skewness− measure of Arnold some extensions. and Groeneveld (1995). With only three parameters there is a one-to-one relation between (the absolute 2. THE ORIGINATORS: FECHNER AND value of) skewness and kurtosis. The conventional LIPPS moment-based measure of kurtosis, β2, ranges from 3 (symmetry) to 3.8692 (the half-normal extreme Gustav Theodor Fechner (1801–1887) is known as asymmetry), hence the distribution is leptokurtic. the founder of psychophysics, the study of the re- Quantiles of the distribution can be conveniently lation between psychological sensation and physi- obtained by scaling the appropriate standard nor- cal stimulus, through his 1860 book Elemente der mal quantiles. For the respective cumulative distri- Psychophysik. Stigler’s (1986, pages 242–254) as- bution functions (CDFs) F (x) and Φ(z) we define sessment of this “landmark” contribution concludes THE TWO-PIECE NORMAL DISTRIBUTION 3 that “at a stroke, Fechner had created a method- the use of the Gaussian distribution to calculate the ology for a new quantitative psychology.” However, probability of an observation falling in a given inter- his final work, Kollektivmasslehre, is devoted more val. The measure of location is the arithmetic mean, generally to the study of mass phenomena and the A, and the measure of dispersion is the mean abso- search for empirical regularities therein, with ex- lute deviation, ε (related to the standard deviation, amples of frequency distributions taken from many in the Gaussian distribution, by ε = σ 2/π). Ta- fields, including aesthetics, anthropology, astron- bles of the standard normal distributionp are not yet omy, botany, meteorology and zoology. In his Fore- available, and his calculations proceed via the error word, Fechner mentions the long gestation period function (see Stigler (1986), pages 246–248, e.g.), of the book, and states its main objective as the and prove to be remarkably accurate. establishment of a generalisation of the Gaussian In previous work Fechner had introduced other law of random errors, to overcome its limitations of “main values” of a frequency distribution, the Zen- symmetric probabilities and relatively small positive tralwert or “central value” C, and the Dichteste and negative deviations from the arithmetic mean. Wert or “densest value” D, subsequently known He also appeals to astronomical and statistical in- in English as the median and the mode. Arguing stitutes to use their mechanical calculation powers that the equality of A, C and D is the exception to produce accurate tables of the Gaussian distri- rather than the rule, he next introduces the Zweis- bution, which he had desperately missed during his paltiges Gauss’sche Gesetz to represent this asym- work on the book.