Biomelrika (1962), 49, 1 and 2, p. 133
Printed in Qreat Britain
133
Ranks and measures
BY M. G. KENDALL
G-E-I-R (U.K.) Ltd.
2. I consider a situation in which we can rank in order of magnitude not only the original
objects but also their differences. For instance, given a class of students ranked on merit
from 1 to 10, we may be able to say that the biggest gap occurs between the fifth and sixth, the
next biggest between the ninth and tenth, and so on. Such a ranking of differences might be:
Student
Ranking of differences
1
2
6
3
3
4
9
5 6 7 8 9
8 1 7 4 5 1
10,]
.
Suppose further than than we rearrange these first differences in order of magnitude and
can rank the differences between them, e.g.
1
2
6
3
1 7
4
5
5
6
2
7
8
8 9
4 3
(2)
We can imagine the process continued until we arrive at a single pair of differences which can
be ranked either 1, 2 or 2, 1. In this whole array of rankings of successive differences we
obviously have a large amount of supplementary information. The question with which I am
concerned is: how can this extra information be used to set up a measure for the original
objects?
3. The procedure proposed may best be illustrated by a numerical example. Column (1)
of Table 1 shows, in order or occurrence, the first eight normal deviates from Quenouille's
(1959) tables of random numbers. In column (2) these are multiplied by 100 and increased
by 100 to remove negative values. Column (3) rearranges them in order of magnitude,
starting with the smallest. Column (4) gives the first differences and column (5) the rank
order of those differences, again starting with the smallest. They are rearranged in order in
column (6) and differences in column (7). Column (8) gives the rankings. In this case two
values in column (7) are tied so the corresponding ranks are split. And so we proceed down
to column (19).
4. The method I propose for reconstituting the original values consists of starting from
column (19) and working backwards. To begin, we need some assumption concerning the
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1. In many situations occurring in psychology we are able to rank a set of objects
according to a criterion, but cannot measure the quality expressed by that criterion. The
desirability of transforming our information in some way from ranks to measures has led to
procedures of various kinds, some of which appear to work satisfactorily in practice but none
of which really touches the fundamental problem. And this is not surprising when we recall
that, even on the assumption that a measure exists, a set of ranks has irretrievably lost much
of the information implicit in a set of variate-values from which the ranks were derived. To
make progress we must supplement the original ranking information in some way or (less
reliably) substitute assumption for additional information.
134
M. G.
Table 1
Columns
A
(1)
-0-84
(2)
16
(3)
1
1-37
237
16
-018
32
15
2
66
4
135
134
382
135
212
312
237
0-34
1
134
5
52
14
14
66
70
75
7
.
6
70
5
102
102
6
4
1
14
5
2
27
9
.
0
13
75
10
312
37
5
.
2
3
.
o
1 9
.
5 10
.
4 13
37
1
1*
1
8
4
1
1
1
3
3
3
.
8
0
.
2
.
5
.
1
2
2
.
3
1
m
2
3
.
t
382
relative magnitudes of the last two differences which are ranked 1, 2. I shall assume them
to be proportional to 1 and 3 units. The reason for this is that if a magnitude is broken at
random into two parts the expected value of the larger part is three times that of the smaller.
Generally, if a magnitude is broken into n parts the expectation of the parts in descending
\
/
- /- —?—
-\
(3)
1. 1
n'n'
Thus our last pair of differences are assumed to have, in units which it is unnecessary to
specify, the values
.
„
...
in that order. Now we require a second assumption to proceed to the previous differences.
The total' length' in the two differences arrayed in (4) is 1 + 3 = 4 units, with an average of 2.
I assume that a piece of this average length precedes the pair in (4), so that the lengths of the
three differences are 2, 2+1, 2 + 1 + 3, namely
2, 3, 6
(5)
and they are in this order since the ranks in column (17) of Table 1 are 1, 2, 3. We now
proceed by similar assumptions. The total number of units in the three numbers of (5) is 11,
with an average of 11/3. The four differences preceding are then
or proportional to
11, 17, 26, 44.
(6)
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2-82
1
4
15
37
3
.
1
3£
1
82
m
-0-99
1
14
52
0-35
(7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) 19)
(4) (5) (6)
Ranks and measures
135
We can work in proportions throughout until we impose a scaling factor at the final
stage.
Now the rankings corresponding to these four quantities, from column (14) of Table 1
are l£, 4, 1|, 3. We therefore average the first two of (6) to obtain 14 for each and rearrange
(6)aS
14, 44,
14, 26.
(7)
The average is 49/2 and, on multiplication by 2 to remove the fraction, the preceding five
values become
49, 77, 165, 193, 245
and on rearrangement according to the ranking of column (11),
165, 49, 245,
193.
(8)
The reader will wish to verify the next stages for himself. Corresponding to columns (8)
and (5) we find
4123, 8748, 4123, 1458, 2228, 6818.
(9)
870, 2157, 1745, 458, 3208, 2526, 2303, total 13,267.
(10)
These numbers, if we have been successful, should be proportional to the differences
between the eight original values. To make a comparison I make the ends of the scale based
on (10)fitthe original set of numbers by identifying end points. The total range of the original
values in column 3 of Table 1 is 381. The range in expression (10) is 13,267. I therefore
multiply the values in (10) by 381/13,267 = 0-028,718 and accumulate from the term unity,
to obtain
Reconstructed values 1 26 88 137 151 243 316 382,)
(11)
Original values
1 16 82 134 135 237 312 382.1
The agreement seems good.
5. The experiment was repeated 24 times, with a sample of 5 and two samples of 10 from
each of the eight tables given by Quenouille, which comprise samples from sundry different
populations. The results for the samples of 10 are summarized in Table 2. The letters
xv ...,zB indicate the eight populations in Quenouille's notation. The agreement on the
whole seems very fair. The most serious discrepancies occur when one end value lies well away
from the others, as in the first set of x3: and this is to be expected.
6. It is not self-evident that the final scaling should be determined by pinning the end
points. Some experiments were carried out to see whether pinning at the second and ninth
or third and eighth points would result in an improvement. There were undoubtedly cases
where this was so, but I did not perceive any general effect which would justify one method
rather than another. Where outliers are suspected, however, I should be inclined to pin at
points inside the range. For example, in the first set of x3 we get these results:
Original values
Reconstructed:
By pinning end points
By pinning at A, B
By pinning at C, D
41
G
43
A
54
69
69
70
75
B
89
D
111
409
41
41
38
59
45
43
99
54
53
151
66
67
161
68
70
175
72
73
202
78
80
250
89
93
319
105
111
409
126
135
The distortion due to the last value is evident. Apart from this the fits are quite good.
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77,
136
M. G. KENDALL
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Ranks and measures
137
7. In actual practice, of course, it will scarcely ever be possible to rank differences
beyond the second; the effort required of the imagination in psychological experiments
would be too great. But this does not destroy the practical utility of the method. Even if
only first differences can be ranked, some estimate of the original measures can be obtained.
In fact, instead of starting with the ranking of two in the proportion 1:3 we can start with
the ranking of seven in the proportions given by expression (3) with n = 7. This may be
rather rough but it is obviously a good deal better than merely operating on the original
ranks as if they were equally spaced measures, or replacing them by normal equivalent
deviates.
9. In the following paper Prof. Daniels examines some theoretical questions prompted by
the forementioned empirical results. I may, in conclusion, mention a further method which
I have not yet had an opportunity of testing in practice. The successive rankings of differences
are equivalent to linear inequalities in the variable on which they are based. For example,
in Table 1 the first ranking is equivalent to
x1<xi<x3...<x8.
(12)
The second ranking (of first differences) is equivalent to
x5-xt
< x2-xl
< Z4-Z3 < . . . < x6-x5
(13)
and so on. We can now pose a problem in linear inequalities. Given the inequalities of
types (12), (13), etc., between what limits must the original variable lie? I hope to examine
this problem on a subsequent occasion.
REFERENCE
QUENOUHXB, M. H.
(1959). Tables of 1000 standardized deviates from certain non-normal distributions.
Biometrika, 46, 178-202.
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8. The foregoing method, or some modification of it, is in my opinion worth a trial as a
method of scaling. As it stands, it separates a set of objects by distances which approximate
to a measure, but it will not locate those objects on a previously determined scale. For
scaling purposes I should recommend mixing the objects with a set of ' markers' whose
position on the scale has already been determined, in whose position defines the scale. The
relative position of the new unmarked objects can then be determined by reference to the
markers.