SEMAINE D’ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 135 
tions of the notions of univariate distribution and bivariate re- 
lationship. 
With reference to Figs. 2a and 3a, let x be the (unspecified) 
molecular weight of sugar. Physical chemistry tells us that 
the sugar molecules are crystals, all of which have the same 
weight, w=3.01 x 107% grams, that is 180 times the atom 
weight of hydrogen. If the weight pn could be measured exactly, 
the situation would be as shown in Fig. 2a. In practice, the 
weighting is subject to observational error, and if the errors 
follow the normal distribution the measurements will be distri- 
buted as shown in Fig. 3a. The observed average x of this 
distribution provides a point estimate of the unknown molecular 
weight i. Next let x be the molecular weight of a polymere, 
say a specific make of nylon. The nylon molecules are bands 
of different length; that is, x is not a specific number, but a 
variable subject to a specific distribution, say as shown in 
Fig. 4a. Here jv denotes the mathematical expectation of the 
distribution, 
(37) 
— 
rt, 
‘ 
‘ 
Distinguishing between the theoretical and the observed distri- 
bution, as illustrated in Fig. sa, the observed mean x gives a 
point estimate of the theoretical mean p. Conceptually, the 
dotted curve represents the distribution of a variable x =x* +¢ 
which is composed of a variable x* with the same distribution 
as in Fig. 4a, and an observation error € which for fixed x* has 
a distribution of the same type as in Fig. 3a. In the present 
illustration it so happens that the molecular distribution can 
only be observed indirectly, since the individual molecules are 
too small for direct observation. Conceptually, we may think 
of the observed distribution as referring to the individual mo- 
lecular weights subject to observational error. 
Comparing the situation in Figs. 2a and 3a with the more 
general situation in Figs. 4a and 5a we note two simple instan- 
ces of attenuated inference: 
21 Wold - pag. 21