BULLETIN 
OF 

THE UNIVERSITY OF TEXAS 
Number 323 
ISSUED SIX TIMES A MONTH 
SCIENTIFIC SERIES No. 27 MARCH 15, 1914 
THE ERROR-RISK OF THE MEDIAN 
COMPARED WITH THAT OF THE 
ARITHMETIC MEAN 

BY 
EDWARD L. DODD, Ph.D. 
Instructor in Mathematics in the University of Texas 


PUBLISHED BY THE UNIVERSITY OF TEXAS, 
AUSTIN, TEXAS 

Entered as second class matter at the postoflicc at Austin, Texas 

411-214-700-:,08& 
BULLETIN 
OF 

THE UNIVERSITY OF TEXAS 
Number 323 
ISSUED SIX TIMES A MONTH 
SCIENTIFIC SERIES No. 27 MARCH 15, 1914 
THE ERROR-RISK OF THE MEDIAN 
COMPARED WITH THAT OF THE 
ARITHMETIC MEAN 

BY 
EDWARD L. DODD, Ph. D. 
lnsrructor in Ma:hematics in the University of Texas 

PUBLISHED BY THE UNIVERSITY OF TEXAS, 
AUSTIN, TEXAS 

Entered ns secor.d class matter at tbc posioffice at Austin, Texas 
Cultivated mind is the guardian g~nius of democracy. • • • • It is the only dictator that freemen acknowledge and the only security that freemen desire. 
President Mirabeau B. Lamar. 
The benefits of education and of useful knowledge, generally difl'used through a community, are essential to the preser­vation of a free government. 
President Sam Houston. 
THE ERROR-RISK OF THE MEDIAN COMPARED WITH 

THAT OF THE ARITHMETIC MEAN. 
BY EDWARD L. DODD. 
Ifseveral leaves * from a tree have been collected, the question arises: What is the typical or representative length of leaf for this tree? Or, if several measurements have been made of the same quantity, the question arises: What shall we accept as an ap­proximation for the "true value" of the unknown magnitude? The "typical value" of the length of a leaf and the "true value" of a magnitude exist usually only as mathematical abstractions. Nevertheless, if we arrange the leaves in the order of their magni­tude-as to length-it appears more reasonable to select the middle leaf as the typical leaf than to select the first and shortest leaf. The length of this middle leaf is called the median of the lengths. On the other hand, it is not so clear that the median is preferable to the average-the arithmetic mean-of the lengths.t In fact, it seems probable that sometimes the arithmetic mean is to be pre­ferred, and sometimes the median. Nevertheless, we may with advantage investigate these two functions of the measurements from the standpoint of general statistical theory. 
In this paper, the Gaussian Probability Law will be used as a criterion. It is not pretended that all distributions follow closely this law in its most simple-symmetric-form. But the Gaussian Law has been commonly accepted as representing the ideal distri­bution, both on account of theoretic considerations and by reason of experimental verifications. For example; an urn contains N balls, of whi.ch M are white. The balls are mixed, and n balls are drawn-where n<N-and then returned to the um. Of these, m are found to be white; and thus m,'n is a "measurement" of the "true value," MIN, which in this case is exactly known. The distribution of the repeated "measurements" about the "true value,'' MIN can be studied experimentally. 
For mathematical treatment, it is assumed that a "true value" or "typical value" or "normal value" exists: let this be denoted by a. The difference, a -m, where m is a measurement, is called the error! of that measurement. The Gaussian Law assumes the existence. of a measure of precision, h, for a set of measurements 
*See King's Elements of Statistical Method, p. 101, Macmillan (1912). 
tFor an elementary discussion of the advantages and disadvantages of the arithmetic mean, the median, the mode, etc., see King, loc. cit. Chap. 
XII. 
tSuch terms as "deviation," "departure," are also used to designate a-m or m-a. 
Bulletin of the Uni-versity of Texas 
made under like circumstances; and then states that the probabil­ity that the error of a measurement will lie between x1 and x 2 is 
(1) 
In the language of "large numbers," this means that when the number, n, of measurements is large, it is to be expected that about pn of these measurements will have errors greater than x 1 and at the same time less than x2• This· law (1) is frequently expressed as follows: The probability that the error of a measure­ment will lie between x and x+dx is 
h -h•x• -;=-e dx. t/ 7r  (2)  
Let  
@ (x) =  2V ,..  Jx -t• e dt.  (3)  
0  

Then from (1) the probability that the error of a measurement will lie between -x and xis @ (hx). 
As preliminary to the treatment of the median, lbt us find the probabilty that: The first error-the error of the first measure­ment-will lie between x and x-t dx; the second error will be less than the first error; and the third error will be greater than the first error. The probability of the first part of this compound event is given by (2). For the moment, suppose that x is positive, and that the first error'is exactly x. Then the probability that the second error will be less than the first error,-i. e., that the second error will lie between-oo and x,-is !+! 0 (hx). The probability that the third error will be greater than the first error is!-!@ (hx). The probability of the compound event is then 
h -h2x•
i V .)1-@ ~(h."l:)]e dx; (4) 
that is, the product of the probabilities of the three components. 
The restriction that x be positive is not essential; since@ (-x)= -® (x). The supposition that the first error is exactly x is a common device, incident to the extension of the law for compound probability-deduced from the consideration of a finite number of possibilities-to the continuum with its infinity of real numbers. We do not deny that such an extension involves some axiom. 
If now 2n+1 measurements are to be made, the probability 
The Error-risk of the 1'11edian 
that: The first error will lie between x and x+dx; the next n errors will be (algebraically) less than the first error; and the last n errors will be greater than the first error, is obtained from (4) by raising t and the bracket each to the nth power. If, however, we simply require that some unspecified one of the 2n+ 1 errors shall lie between x and x+dx, and that of the 2n remaining errors 
just n errors shall be less than x, the coefficient (2n+ 1) ~~1dis 
also needed. 
Suppose now that the error of the median of 2n+1 measure­ments is x. Then the errors of n of the measurements must be (algebraically) less than or equal to x; and the errors of the remain­ing n measurements must be greater than or equal to x. The probability that the error of the median of 2n+1 measurements will lie between x 1 and x 2 is, then, 

If we set  
N-(2n+l)! · t=hx-t =hx · t =hx · 4D(n !) 2 1 1 1 l 1 2_ 2 1  (6)  
this becomes  
N t,-J [1-0-r1-;­ n -t• 2 (t)] e dt.  (7)  

t, 
An interesting verificat.ion of (5) and (7) consists in setting x1 = ·-· oo, x2_=+ oo; and introducing the change of variable, 
2 -t2 
u= @ (t), du=v ,.. e dt. 
Then (7) becomes 
NJ1 (1-u2 )n du. (8) 0 
By the substitution, u=sin t/>, this can be shownt to be equal to unity, the symbol for certainty. We shall now make use of the conception of the probable value. To illustrate: If a gambler is to receive one dollar if he throws an 
tPeirce-A Short Table of Integrals, p. 62. 
Bulletin of the University of Texas 
ace with a die, two dollars if he throws a two-spot, etc., his expecta­tion on a single throw is defined as 
t(l)+t(2) +t(3)+t(4) +t(5)+t(6)=3! 
dollars; and this is also called the probahle value of his intake. It is obtained by multiplying each possible prize by its probability-in this case, i-and taking the sum of the products. It may be roughly described as the expected average intake. The gambler's "expectation" in sixty throws is 60X3!=210 dollars; and this is exactly what he will receive if he throws an ace ten times, and each other spot just ten times. 
Now let the absolute or positive value of an error, x, be called the absolute error, and be denoted by Ix!-Then, in accordance with the usual generalization, the probable value of the absolute error of the median is obtained by multiplying the integrand in (5) by lxl, and taking x1 =-oo, x~=oo; or, what is equivalent, multiplying the integrand by 2x, and taking x1 =0, x2 =oo. 
We can obtain directly a theorem comparing the probable value of the absolute error of the median with that of the arithmetic mean; but we shall obtain it as a corollary to a theorem on error­risk. 
In dealing with error-risk-"Fehlerrisiko"-Czuber* intro­duces a risk-function with the following characteristics: 
v(-x)=v(x); and v(x')>v(x) if lx'I> Ix!. (9) 
It is to be supposed also that v is not negative, and that 
-h•x• 
v(x)e is integrable from 0 to oc. The most simple risk-
functions are, perhaps, 
jxj, x2, jx3 j, x4, etc. 
By the error-risk is meant the probable value of v(x), where xis the error of a measurement or of a function of the measurements. Hence, from (5) and (6),. the error-risk of the median of 2n+ 1 measurements is 
2N Joo (X) -x•
n
Pi=---= v -[l-® 2 (x)]e dx. (10)
JI 71" h 
0 
This is obtained by first making the substitution, hx=t; and then, t--=x. Now, the arithmetic mean of 2n+ 1 measurements; each subject 
*W ahrscheinlichkeitsrechnung I. p. 267. 
The Error-risk of the Median 
to the Gaussian Law (1) with measure of precision, h, is itselft subject to the Gaussian Law with measure of precision, 

(11) 
Hence, the error-risk of the arithmetic mean of 2n +1 measure­ments is 
00
2v2n+1J (x) -(2n+l)x1 
P= v " v h e dx. (12) 
0 
It will now be shown that P is less than P 1 for each (positive integral) value of n. In the technical sense, as used here, the "risk" of error in accepting the arithmetic mean is, then, less than in accepting the median. In other words, it is to be expected that the error of the a.rithmetic mean will be numerically less than that of the median in Gaussian distributions. 
To prove this, consider the two curves, suggested by (12) and (10), viz., 2y2n+l -(2n +l)x• y=F(x)= V" e , (13) 
2N n -x• y=f(x) = ..;;[1-@ 2 (x)] e (14) 
In what follows, we shall consider only values of x>O or x=O. We sl;iall first prove that F (O)>f(O). Stirling's formula, in closed form*, is 
e 
n -n+­
(15)
n!=y2"n n e 12n, 0<8<1. 
Then, from (6), 
9' 
2n -2n+­
N = (2n+1) (2n!) (2n+1) ( 1/ 4;ii(2n) e 24n 
4 n (n!) 2 n 2n -2n;--e 
4 (21rn)n e Sn 

From this, it follows that 
1 1 
-6nJ2 N J2n+l 24n
e -< < __ e 
11" v2n+1 'll"n 
tCzuber, loc. cit. p. 262, gives a more general statement of this principle. *Cesaro-Corso di analisi algebrica, pp. 270, 480. 
Bulletin of the University of Texas 
And hence, when n > 1, 
i< ~<1, (16)
I/ 2n+l 
which is, indeed, also satisfied when n=1. Hence, F(O) >f(O); since ® (O)=O. 
We shall next prove that F(x) <f(x) when x>2. In other words, beyond x=2 the curve, y=F(x), lies wholly below the curve, y=f(x). Since ® ( oo) = 1, 
2 00 -x•
1-® (x) =-J e dx.
v-; 
x Now, if x>2, 
-xr 
8xe  <1;  
and thus,  2Joo  00J(4xe-x") e-x"dx< e -x"dx.  
x  x  
-2x•  

But the left member is equal to 2e ; and hence, 
-2x" 1-® (x)>2e 
n n -2nx2 
[1-® 2(x)] >2 e 
This, with (16), proves that F(x)<f(x) when x>2. 
Hence, the curves (13) and (14) must intersect at least once-as stated before, we are not considering negative values of x-and, if they intersect twice, they must intersect at least three times. By the indirect method, we shall prove that the curves cannot intersect in three points; and hence, they intersect but once. Suppose, then, that there are three points of intersection. And let 
n N
D(x)=F(x)-f(x); c ./ , c>O. 
(17)
v 2n+l Then 2 -x• --{ -2nx• nD(x)=,-=e p2n+l e -c [1­
1' 1r 
One factor of D (x) is -2x• 
g(x)=e -c[l-® 2 (x)]; 
and the product of the remaining factors is positive. Then, since D(x) has three positive roots, g(x) also has three positive roots; and 
The .E'rror-risk of the Median 
-2x1 2 -x• g'(x) = -4xe +2c @ (x) p-;e 
has two positive roots, by Rolle's Theorem. Thus, -x2 c
G(x)= -xe + -= @ (x)
v ... 
has two positive roots. But G(O) is also zero. And hence 
-x2 2c -x• 
G'(x)=(2x2 -l)e +-e 
7r 
has two positive roots. But 2x' _ 1 +2c 
7r 
can not be zero for more than one positive value of x. Thus, the­supposition that (13) and (14) intersect in more than one (positive) point, leads to an impossibility. 
Let b be the abscissa of the point of intersection, and let A be the area between the two curves (and the Y axis) from x=O to x=b. Tb.en, the area between the two curves from x=b to x=oo is also A; since the total area beneath each curve is unity­see (8). Now, from (10), (12), (13), (14), (17), 
P f00 F(x)v{~) dx=fbD(x)v{~) dx+fbf(x)v{1)dx+ 
[ 00 F(x)v{~) dx; (18) 
(X) b (X) 
P1 = [ f(x)v{~)dx=f f(x)v{~)dx+[ F(x)v{~)dx+ 



[ 00 -D(x)v{~)dx. (19) 
But the first integral on the right in (18) is less than Av(~); 
since, by (9), vis an increasing function of x. The last two inte­grals in (18) are the same as the first two integrals on the right in (19). In the last integral of (19), the function -D(x) is posi­
Bulletin of the University of Texas 
tive*; since F(x)<f(x)  when x>b.  This integral is greater than  
Av(~)  Hence  

 (20)  

THEOREM 
The error-risk of the median of an odd numbert of measurements, each subject to the Gaussian Law (1), is greater than the error-risk of the arithmetic mean of these measurements. 
COROLLARY 
The probable value of the absolute error of the medfon of an odd numbert of measurements, each subject to the Gmtssian Law (1), is greater than that of the arithmetic mean of these measurements. 
This corollary is also evident from mechanical considerations,­when it has been proved that F(O) >f(O); and that the curves, y=F(x) and y=f(x), intersect but once. For the two probable values, P and P 1 are simply the abscissas of the centers of gravity ()f the areas beneath these two curves, respectively. 
Now a center of gravity problem may be looked upon as a problem of finding an average,-involving, perhaps, a passage to a limit. And there is enough resemblance between a probable value and an average to lead to a suspicion that a method of comparison based upon probable value might favor the arithmetic mean, M, above all other functions of the measurements. That this is not indeed the case would appear from the fact that there are func­tions** with an error-risk smaller than that of the arithmetic mean. And futhermore, these comparisons based upon probable value are in harmony-so far as they go-with comparisons obtained by another methodtt. 
The median is not one of the functions considered by Czubert in his treatment of error-risk. For, although the median is a continuous function of its arguments-the measurements-the first partial derivarives do not always exist; and thus there are 
*It might be zero at just one point, so far as has been proved; but this does not affect the argument. tThe degenerate case, n = 1, is not here considered. **As found in a former investigation. See Monatshefte fuer Mathematik 
und Physik, 1913, p. 270. For example: ( 1-n\)M, when n is large 
enough,-using (8) p. 270. 
ttAnnals of Mathematics, June 1913, pp. 186-198. 
tLoc. cit. p. 275. 
The Error-risk of the llfedian 
points about which the median can not be given a Taylor's develop­ment. To illustrate: Suppose the measurements are x, y, and z; and that the median *is F(x, y, z); and suppose x< z. If, now, x<y<z, then the partial derivative, Fy'(x, y, z)=l. But, if y=x, the forward Fy' is equal to 1; whereas, the backward Fy' is equal to 0. And thus, at this point, Fy' itself does not exist. Czuber considers functions that can be given a Taylor's development. 
The attempt has been made to show that the Gaussian Probabil­ity Law* is a logical consequence of the "Principle," that the arithmetic mean is the "most probable value" of the unknown true value. But Bertrandt gives an example to show that the Law and "Principle" are not strictly compatible. What function of the measurements the Gaussian Law endorses above all other functions is not at present known. And it may well be that there is no such function. 
The theorem of this paper does not recommend the use of the arithmetic mean in preference to the median under all circum­stances; but merely when it is assumed that the distribution is a Gaussian distribution. 
But, in view of this theorem, if there is any reason for supposing -in a particular case--that the median gives a more satisfactory result than the arithmetic mean, this is also a reason for supposing that the distribution is not Gaussian. And thus," any formula for "probable error" or like expression deduced from the assumption of the Gaussian Law should be viewed with considerable suspicion, so far as its application to the particular case is concerned. 
•This Fis entirely distinct from that used in (13). 
tCalcul des Probabilites (1889) p. 180.