THE AMERICAN STATISTICIAN
, VOL. , NO. , –
http://dx.doi.org/./..HISTORY CORNER

On Galton’s Change From “Reversion” to “Regression”
Prakash Gorroochurn

ABSTRACTARTICLE HISTORY

Galton’s first work on regression probably led him to think of it as a unidirectional, genetic process, which hecalled “reversion.” A subsequent experiment on family heights made him realize that the phenomenon wassymmetric and nongenetic. Galton then abandoned “reversion” in favor of “regression.” Final confirmation
was provided through Dickson’s mathematical analysis and Galton’s examination of height data on brothers.

Received April 
Revised August 1. Introduction
Regression, as we know it today, was born from Galton’s investigations into the laws of heredity. The phenomenon that Galton
discovered is best described in his own words:…offspring did not tend to resemble their parent seeds in size, but tobe always more mediocre than they—to be smaller than the parents,
if the parents were large; to be larger than the parents, if the parents
were very small … (Galton 1886)

That Galton came to this conclusion almost single-handedly
and not by drawing on the contributions from his predecessors is testimony to his genius. The various experiments andanalyses that Galton performed before he reached his conclusion have been well described in works such as Cowan (1972);
MacKenzie (1978); Porter (1986); Stigler (1986, 1989); and Bulmer (2003). However, what is often not properly discussed isthat Galton at first very probably did not understand regression
as we know it today. He first called the phenomenon “reversion” (indeed the symbol r was first used by Galton to signify“the coefficient of reversion” (Pearson 1930, p. 9), which wasa genetic process well known to both him and his contemporaries. One of his first discoveries was not that there was a regression effect, but rather that the reversion phenomenon he had
observed and had assumed would occur was operating in a linear fashion. Galton also thought the phenomenon was a unidirectional process operating on offspring from remote ancestors(Gorroochurn to appear, Chap. 2). The realization that something other than a unidirectional genetic process was going on,
however, soon came about when he found that reversion was
also occurring on parents from their offspring. At this stage, Galton made the decision to change “reversion” to “regression.” Galton confirmed his hypothesis through the mathematical analysis of J. Hamilton Dickson and later through the examination ofheight data on brothers. The phenomenon was not genetic reversion as he had at first thought, but a nongenetic, purely statisticalphenomenon that could operate in either direction.

2. The 1877 Paper: Reversion or “ATAVISM”
Galton’s (1877) groundbreaking paper, “Typical Laws of
Heredity” dealt with a problem that had preoccupied him
CONTACT Prakash Gorroochurn©  American Statistical Associationpg@columbia.edu

KEYWORDS

Ancestral type; Atavism;
Imperfect correlation

for a while, indeed since his 1869 book Hereditary Genius (Galton 1869): Why do the characteristics (mean and variance) ofa hereditary attribute (such as height) from an isolated humanpopulation remain constant from generation to generation? To
explain the constancy in attributes, Galton invoked reversion,
also known as atavism, which is the genetic process by which anindividual resembles a grandparent or more distant ancestor
with respect to some trait not possessed by the parents. This canhappen, for example, if a recessive and previously suppressed
trait reappears through the combination of two recessive alleles
in a genotype. Alternatively, the process of recombination cangive rise to a unique constellation of genes resulting in a longsuppressed character to reappear (these two mechanisms were
not known to Galton as Mendelism was yet to be rediscoveredin 1900). Atavism is thus reversion to ancestral type and was wellknown by Galton’s contemporaries, including Darwin (1859, p.
14), who, in fact, first proposed it. This genetic process is quite
different from the purely statistical phenomenon that Galtonsoon discovered and at first identified with reversion.
There is undeniable evidence that Galton believed thatatavism was the process that would revert offspring’s traits to
those of their distant ancestors. Thus, back in 1865, he made thefollowing statement:
Lastly, though the talent and character of both of the parents might,
in any particular case, be of a remarkably noble order, and thoroughly congenial, yet they would necessarily have such mongrel
antecedents that it would be absurd to expect their children to invariably equal them in their natural endowments. The law of atavism
prevents it. (Galton 1865, p. 319).

Galton’s (1877) in his paper explained that he resorted toexperiments with sweet peas to answer his questions. He sorted
a large number of sweet pea seeds into seven equally spaced size
(weight) classes and sent each of his friends seven packets, eachcontaining 10 seeds of a given class size. The seeds from the offspring were then collected and sent back to Galton. From hisanalysis of the seed results, Galton made the following two key
conclusions:
1. For a given parental class size, the size of the filial seeds
was normally distributed, with the same probable error
e p within each class (i.e., the same family variability).

Associate Professor, Department of Biostatistics, Columbia University, New York, NY .

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P. GORROOCHURN

2. Reversion followed “the simplest possible law,” being alinear function of the deviation from the grand mean
(M). Thus, if the parent size in a given class has mean
M + ke p (k = 0, ± 1, ± 2, ± 3) , then the corresponding filial size will have mean M + kρ e p , where, as weshall see below, ρ (0 < ρ < 1) is the fractional coefficient of reversion.
Notice that, in the above, Galton did not say that he has discovered reversion, the latter genetic process having already been
assumed to be operational; rather Galton stated that it was linear.
In an Appendix to the 1877 paper, Galton gave the algebraic
conditions necessary for stability in population variability interms of the modulus c of a distribution (the modulus√of a normal distribution was historically used to represent σ 2 ). Galton first considered the case of two parents each of whom could
be productive, but later turned to the case of simple descent. It
is the latter case that we shall describe here. First, Galton wrote
the distribution of the “amount of deviation” x in a present population as 2
x1
y = √ exp − 2 .
cc πSecond, “reversion is expressed by a simple fractional coefficient of the deviation,” which we shall denote by ρ (0 < ρ < 1)
(Galton himself used the symbol r, but we shall reserve the latter for the sample correlation coefficient). Then, in the “reverted
parentages,”1x2
y = √ exp − 2 2 .
ρ cρc πGalton then denoted the modulus of the present populationby c1 and that of the “reverted parentages” by c2 , so thatc2 = ρc1 .

(1)

The next step for Galton was to consider the variation of thenumber of progeny for a given parental class, that is, the family
variability.
Family variability was shown by experiment to follow the lawof deviation, its modulus, which we will write as v, being thesame for all classes. Therefore, the amount of deviation of anyone of the offspring from the mean of his race is due to the
combination of two influences—the deviation of his “reverted”
parentage and his own family variability; both of which follow
the law of deviation. This is obviously an instance of the wellknown law of the “sum of two fallible measures” (Airy, “Theoryof Errors,” §43 (Galton’s footnote) (Galton 1877, p. 533).
Denoting the modulus of the family variability by v , Galton
wrote the above law as
c42 = c22 + v 2 ,(2)

where c42 is the overall modulus of the progeny. Combining
Equations (1) and (2),c42 = ρ 2 c12 + v 2 .
Now, for variability to remain constant across generations, weneed c1 = c4 , which implies
c12 =

v2
1 − ρ2(3)(ibidem). It should be noted that Equation (3) expresses thefact that the variability across generations remains constant due
to the balancing forces of family variability (which tends toincrease spread) and reversion (which tends to reduce spread).
In sum, Galton’s (1877) paper is groundbreaking but still not
fully satisfactory in the sense of projecting a clear and accurate view of regression. Galton’s use of reversion to signify “thetendency of that ideal mean filial type to depart from the parent type, reverting towards what may be roughly and perhaps
fairly described as the average ancestral type” shows that, at thisstage, he thought of the process as being both unidirectional andgenetic. To his amazement, the contrary turned out to be true,
and this was to be shown in his 1885 presidential address to the
Anthropology Section of the British Association (Galton 1885a).
This is also where the term “regression” was first used by Galton,
having realized that something else than atavism was going on.
We now turn our attention to this lecture.

3. The 1885 Presidential Lecture: Regression’s First
Appearance
In 1885, Galton was to make another major breakthrough following his initial 1877 paper on reversion. In the presidential
lecture to the Anthropology Section of the British Association,
Galton admitted the following:…I was then [in 1877] blind to what I now perceive to be the simple
explanation of the phenomenon … (Galton 1885a, p. 507).

Having now realized an important fact he had been “blindto,” Galton was poised to make a departure from his original, lessaccurate “reversion” to the new, more accurate “regression.” Hethus stated that his previous experiments on sweet peas showedthat “the mean filial regression (author’s italics) toward mediocrity was directly proportional to the parental deviation fromit.” Moreover, he now not only showed that regression is symmetric in nature but also gave the correct mechanism for it. (Galton was correct in explaining regression from one generation tothe other. However, Galton also made an argument for perpetual
regression, which turned out to be erroneous. For more details,
see Bulmer 2003, p. 285).
Galton explained that he had taken pains to collect theheights of parents and adult children in 205 families. He thenmultiplied the height of each female parent by 1.08 so as tomake the male and female parents comparable and “no objection grounded on the sexual difference of stature need be raised.”
He then took the average for each pair of parents to obtain a“mid-parent” height. From these, Galton calculated the medianfor both mid-parents (X) and offspring (Y) as 68.25 inch andthe probable errors as e p,X = 1.2 and e p,Y = 1.7 , respectively.
(The probable error of a random variable X is simply its semiinterquartile range, i.e., for a symmetric distribution the value
ep such that Pr{|X-m| < ep } = 0.5. At one time, it was a popularmeasure of variability but was replaced by the standard deviation, a term coined by Pearson (1894). It can be shown that,when X is normally distributed, ep = 0.675 SD.) Next, for each
mid-parental height, Galton calculated the median height of
the adult children and plotted the latter median heights against
the mid-parental heights. The line is fitted by inspection, andTHE AMERICAN STATISTICIAN: HISTORY CORNER

229

resembles that of pouring a uniform proportion of pure water into a
vessel of wine. It dilutes the wine to a constant fraction of its original
alcoholic strength, whatever that strength may have been (Galton
1885a, p. 508).

Figure . Galton’s two regression lines (taken from Galton b).

Galton obtains a slope of 2/3. Galton was now ready to define
his law of regression for these data:
It is that the height-deviate of the offspring is, on the average, twothirds of the height-deviate of its mid-parentage (Galton 1885a, p.
508).

In modern notation, we can write this as2(X − M)
3
2
1
E (Y |X ) = X + M,
3
3

E (Y |X ) − M =where the overall population mean M has been assumed to beconstant.
Galton now took the next step that he unfortunately had not
ventured into in his sweet pea experiments. Having plotted offspring median heights (Y) against the mid-parental heights (X),
Galton next plotted the median mid-parental heights against theoffspring heights, and discovered that a similar reversion effectacts on the mid-parental heights (see Figure 1).
That is, not only were offspring more “mediocre” than their
mid-parents but mid-parents were also more “mediocre” thantheir offspring! This single observation is enough to put a serious
dent in the hypothesis that genetic reversion to ancestral types
was the process responsible for the observations that Galton had
initially made. Galton correctly assessed the new discovery andgave the correct explanation for the regression effect:
The explanation of it is as follows. The child inherits partly from his
parents, partly from his ancestry. Speaking generally, the further his
genealogy goes back, the more numerous and varied will his ancestry
become, until they cease to differ from any equally numerous sample taken at haphazard from the race at large. Their mean stature
will then be the same as that of the race; in other words, it will be
mediocre. Or, to put the same fact into another form, the most probable value of the mid-ancestral deviates in any remote generation iszero.
For the moment let us confine our attention to the remote ancestry and to the mid-parentages, and ignore the intermediate generations. The combination of the zero of the ancestry with the deviate of
the mid-parentage, is that of nothing with something, and the result

Indeed, the above explanation is one of the most intuitive
ways of understanding the regression effect: In general, supposea first measurement X is made on a given subject, followed bya second measurement Y. Assume X is exceptionally high. Aslong as X and Y are imperfectly correlated, X can be thought tobe made up of two components:
1. The first component, which is usually extreme and is
expected to remain extreme.
2. The second component, which is not extreme and is
expected to remain near the center of the distribution.
The first measurement X is extreme because both components are high. However, for the second measurement Y, the firstcomponent is expected to remain high, but the second component is expected to be near the center. Hence, the average valueof Y will be less extreme than X and closer to the center of the
distribution (e.g., Wallis and Roberts 1956, p. 61; Stigler 1997).
Notice that the above explanation does not require any biological, economic, or other force to be present for regression
to occur. The regression effect (or regression to the mean) isa purely statistical artifact arising from imperfect correlation
between X and Y (of course, the concept of correlation was notknown to Galton when he first discovered the regression effect).
As such, it is also symmetric in the sense that the same reasoning as above can made to argue that, for a given extreme Y, thevalue of X is expected to be less extreme and closer to the center.

4. Final Confirmation: Dickson’s Analysis and Data on
Brothers
The irrefutable confirmation of Galton’s hypothesis on the symmetric and purely statistical nature of regression was provided
by the mathematician J. Hamilton Dickson and Galton’s examination of height data on brothers. Galton used a sheet of squared
paper and entered frequencies on it (Figure 2). He noticed that,when identical values (total values of each cell) were joined byline segments, a series of concentric and similar ellipses wereformed (e.g., see Friendly and Denis 2005). Galton then reported
the observations he made:
Their common center lay at the intersection of the vertical and horizontal lines that corresponded to 68.25 inches. Their axes weresimilarly inclined. The points where each ellipse in succession wastouched by a horizontal tangent, lay in a straight line inclined to thevertical in the ratio of 2/3; those where they were touched by a vertical tangent lay in a straight line inclined to the horizontal in theratio of 1/3. These ratios confirm the values of average regression
already obtained by a different method, of 2/3 from mid-parent tooffspring, and of 1/3 from offspring to mid-parent, because it will
be obvious on studying [Figure 2] that the point where each horizontal line in succession is touched by an ellipse, the greatest value
in that line must appear at the point of contact. The same is true inrespect to the vertical lines. These and other relations were evidently
a subject for mathematical analysis and verification. (Galton 1885b,
p. 255)

These observations clearly leaned in favor of Galton’s hypothesis that regression occurred both forwards and backwards, anddid not require a genetic force to be operational. However, an

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P. GORROOCHURN

Figure . Ellipse generated by Galton by joining points with the same frequency.

irrefutable confirmation needed a proper mathematical analysis, a task that Galton thought was beyond his analytical skills.
Therefore, he solicited the help of the able mathematician J.
Hamilton Dickson. In modern mathematical language (this isexactly how the problem was phrased: “A point P is capable ofmoving along a straight line P’OP, making an angle tan−1 2/3
with the axis of y, which is drawn through O the mean positionof P; the probable error of the projection of P on Oy is 1.22 inch:
another point p, whose mean position at any time is P, is capable of moving from P parallel to the axis of x (rectangular coordinates) with a probable error of 1.50 inch. To discuss the ‘surface of frequency of p’ (Galton 1886), Dickson was provided with2),
the information that Y ∼ N(0, σY2 ) and X|Y ∼ N(βX|Y y, σX|Yand was asked the following questions:
1. What is the joint density of (X, Y ) , and what is the shapeof the contours of equal probability density?
2. How can the regression coefficient βY |X be calculated?
3. What is the density of Y given X?
4. What is the relationship between βY |X and βX|Y ?
Dickson answered each of the above questions without muchtrouble, and the solution was published as an Appendix to Galton’s (1886) paper “Family Likeliness in Stature.” In modern
notation, the joint density of X and Y is

To obtain the contours of equal probability, Dickson sets theexpression in the above exponent to a constant (say K):2
x − βX|Y y
y2
+
= K,
2
σY2
σX|Y

which is the equation of a set of ellipses.
To obtain the required regression coefficient βY |X , firstEquation (5) is differentiated:x − βX|Y y 1 − βX|Y dy/dx
ydy/dx
+
= 0,
2
σY2
σX|Y
so thaty
σY2 dy(x − βX|Y y)(dx − βX|Y dy)
= 0.
2
σX|Y



x − βX|Y y βX|Y
y
−
= 0,
2
σY2
σX|Y
or
y=

(4)

(Pearson (1921) expressed his puzzle as to why Galton did nothimself derive the joint density of X and Y since he alreadyknew both fY (y) and fY|X (x|y). However, it is unlikely that Galton
thought in terms of conditional and marginal distributions.)+

By setting the coefficient of dy to zero, tangents to the ellipse
in Equation (5) parallel to the y-axis can be obtained and theseintersect the ellipse at points lying on the line OM (see Figure 2)
with the following equation:

 
fXY x, y

 
2 

 
 
x − βX|Y yy2= fY y fX|Y x|y ∝ exp −
+
.
2
2σY2
2σX|Y

(5)

βX|Y σY2x.
2 + β2 σ 2
σX|Y
X|Y Y

Thus,
βY |X =

βX|Y σY2.
2 + β2 σ 2
σX|Y
X|Y Y

(6)

THE AMERICAN STATISTICIAN: HISTORY CORNER

 
To obtain the conditional density fY |X y|x with the aid of
Equation (6), the exponent in Equation (4) can be rewritten as2x − βX|Y y
y2
+
2
2σY2
2σX|Y
x2 +

2
2
σY2 + σX|Y
βX|Y

σY2

=

y2 −

2xβX|Yy22 /σ 2
βX|Y + σX|Y
Y

2
2σX|Y
2

y − βY |X x
x2
+ 2

=
,
2 / 1−β2σX|Y βY |X /βX|Y
2σX|YX|Y βY |X

which gives the two results:X ∼ N 0,2σX|Y

1 − βX|Y βY |X

2
Y |X ∼ N βY |X x, σX|Y

correlation between two measurements. Galton then decided tochange “reversion” to “regression.” Of course, Galton cannot be
blamed for at first having been “blind” to the correct explanation of the regression phenomenon, as he himself admitted. Galton was a pioneer in the true sense of the word and, as such, itwould be quite unfair to expect anything more than what he hadalready achieved.

References

,

βY |X
.
βX|Y

231

(7)

(It can be shown that βY/X βX/Y = ρ 2 , the square of the correlation coefficient between X and Y.)
Dickson also gave the relationship between βY |X and βX|Y
through Equation (6).
Galton was clearly elated when he received Dickson’s analysis, since it confirmed all his empirical results. As Galton had
suspected, with only the three pieces of information σY2 , βX|Y ,2and σX|Y(together with the normal distribution assumption),
the elliptical contours he had observed could be constructed.
Moreover, given that there was linear regression of X on Y, Dickson had not only shown that there was also linear regression of
Y on X but also had been able to deduce coefficient for regression βY |X . These were powerful results confirming Galton’s
hypothesis that regression was both symmetrical and intrinsically statistical (e.g., Denis 2001; Maraun, Gabriel, and Martin
2011).
It is also very likely that Galton’s appreciation of the consequence of symmetry was to be further consolidated, in addition
to Dickson’s analysis, through his later examination of the dataon brothers (Galton 1889, p. 210; Stigler 1986, p. 290). The data
were obtained from returns of 295 families and comprised 783
brothers in all. From these, Galton constructed a table showing the distributions of heights of brothers of men with a given
height. The resulting table turned out to be symmetric, reflecting
the symmetric nature of regression (although, as Stigler (1986,
p. 293) has pointed out, the degree of symmetry was exaggerated
due to Galton erroneously counting each pair of brothers twice).

5. Conclusions
Galton’s first encounter with the regression phenomenon led
him to believe that reversion or atavism was really the process
operating. This process is both unidirectional and genetic. The
subsequent realization that reversion was actually symmetricmade him realize that something else was going on, namely,a purely statistical phenomenon arising from the imperfect

Bulmer, M. (2003), Francis Galton: Pioneer of Heredity and Biometry, Baltimore, MD: Johns Hopkins Press. [227,228]
Cowan, R. S. (1972), “Francis Galton’s Statistical Ideas: The Influence ofEugenics,” Isis, 63, 509–528. [227]Darwin, C. (1859), On the Origin of Species by Means of Natural Selection,
London: John Murray. [227]
Denis, D. (2001), “The Origins of Correlation and Regression: Francis Galton or Auguste Bravais and the Error Theorists?” History and Philosophy of Psychology Bulletin, 13, 36–44. [231]
Friendly, M., and Denis, D. (2005), “The Early Origins and Developmentof the Scatterplot,” Journal of the History of the Behavioral Sciences, 41,
103–130. [229]
Galton, F. (1865), “Hereditary Talent and Character,” Macmillan’s Magazine,
12, 157–166 (Part I), 318–327 (Part II). [227]——— (1869), Hereditary Genius, London: Macmillan (Reprinted 1979,
London: Friedmann). [227]——— (1877), “Typical Laws of Heredity,” Nature 15, 492–495, 512–514,
532–533. (Also in Proceedings of the Royal Institution 8, 282–301).
[227,228]——— (1885a), “Presidential Address, Section H, Anthropology,” Nature,32, 507–510 (Also published in (1885) British Association Reports 55,
1206–1214). [228,229]——— (1885b), “Regression Towards Mediocrity in Hereditary Stature,”
Journal of the Anthropological Institute, 15, 246–263. [229]——— (1886), “Family Likeness in Stature,” Proceedings of the Royal Societyof London, 40, 42–73. (Appendix by J. D. Hamilton Dickson, 63–66).
[227,230]——— (1889), Natural Inheritance, London: Macmillan. [231]
Gorroochurn, P. (to appear), Classic Topics on the History of Modern Mathematical Statistics: From Laplace to Modern Recent Times, Hoboken, NJ:
Wiley. [227]
MacKenzie, D. A. (1978), Statistics in Britain, 1865–1930, Edinburgh, UK:
Edinburgh University Press. [227]
Maraun, M. D. , Gabriel, S., and Martin, J. (2011), “The Mythologizationof Regression Towards the Mean,” Theory & Psychology, 21, 762–784.
[231]
Pearson, K. (1894), “Contributions to the Mathematical Theory of Evolution,” Philosophical Transactions of the Royal Society of London, 185, 71–110. [228]——— (1921), “Notes on the History of Correlation,” Biometrika, 13, 25–45.
[230]——— (1930), The Life, Letters and Labours of Francis Galton (Part IIIA),
Cambridge, UK: Cambridge University Press. [227]Porter, T. M. (1986), The Rise of Statistical Thinking, 1820–1900, Princeton,
NJ: Princeton University Press. [227]
Stigler, S. M. (1986), The History of Statistics: The Measurement ofUncertainty Before 1900, Cambridge, MA: Harvard University Press.
[227,231]——— (1989), “Francis Galton’s Account of the Invention of Correlation,”
Statistical Science, 4, 73–79. [227]——— (1997), “Regression Towards the Mean, Historically Considered,”Statistical Methods in Medical Research, 6, 103–114. [229]
Wallis, W. A. , and Roberts, H. V. (1956), Statistics: A New Approach, Glencoe, IL: The Free Press. [229]