THE LOGICAL STATUS OF MENTAL FACTORS 127

years, as in estimating development or achievements at different
ages (e.g. [41], pp. 273, 300).

(ii) Graded scales. In these, instead of being all of equal diffi-
culty, the test-items are arranged in order of increasing difficulty,
proceeding by approximately equal steps. The comparative diffi-
culty of each test-item is usually determined by the proportion o£
examinees giving correct responses (e.g. [41], p. 138, etc.)* With
this and the preceding type of test the initial figures will not
necessarily space out the positions on the scale at equal intervals.
In psychology, as in physics, the requirements of equality may
oblige us to take, not the experimental figures themselves, but some
function of them that will yield more plausible results—that will lead,
for example, to laws or correlations expressible in linear form.

(iii) Impressionistic methods : qualitative scales with subjectively
determined units. In these the equality of units is decided intro-
spectively. Consciously or unconsciously, the observer relies on his
judgment of just-perceptible differences or of equal-appearing
intervals, much as he does in estimating time or length when no
subdivisions on the dial or measuring-rod are visible. For more
accurate results the observers and the observations will be multi-
plied, and the figures adjusted by statistical treatment—e.g. by
averaging, or by invoking the familiar principle that * differences
noticed equally often are equal.'

(iv) Analytic methods : qualitative scales with objectively
determined units. The total product is analysed into a number of
separable or distinguishable elements (for example, the component
test-processes in the booklets for testing intelligence, or what are
supposed to be the essential elements of good writing in tests of
English composition). The mark for the whole is then obtained as
a weighted or unweighted sum of the several parts. Except in the
simplest cases, this in itself really entails a preliminary factor-
analysis ; but, unless a definite experiment is first carried out (such
as that described below), there is no guarantee that the intervals
denoted by the same figure in different parts of the scale will really
be equal (cf. [41], pp. 308, 331 ; and [134]).

(2) Operations.—Both the construction of such scales, and
their application to new examinees, call for some principle
whereby we may determine equality, i.e. some clearly
defined operation of * matching ' by means of which we can
decide whether the fundamental differences (or * intervals')
are themselves different or not. What is meant by saying
that two such distances are interchangeable ? Physical