THE
VARIANCE FOR MULTIPLE-CALLING
By J. G. PRATT
In conducting tests of ESP, it is sometimes not convenient to provide a separate target for each call. For example, a classroom test may be conducted with much greater economy of effort if all the students are permitted to call the same pack of ESP symbols instead of having a separate order of targets for each one. Greville (Annals of Mathematical Statistics, 1944, Vol. XV, No. 4) has worked out the statistical method for obtaining the exact theoretical variance for such a case of multiple calling. His publication is not readily available to research workers, and the generalized statement of the problem is in symbolic mathematical terms which relatively few investigators are able to apply to a specific set of data without the help of a statistician. The purpose of this note is not to explain the underlying theory of this method but to illustrate its use as applied to multiple calling of ESP targets and to provide simple directions for making the computations.
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Two
target situations which are commonly used may be considered.
(1)
The
targets are chosen at random, so that the probability of any one of the five
symbols being the target is independent of the other targets.
(2)
The
targets are chosen by taking the order of symbols in a randomly selected
permutation of a closed ESP pack, i.e., 25 cards consisting of 5 of each of the
5 symbols.
For both
( 1 ) and (2), chance expectation of correct calls is 20% of the total number
of calls. This is true for both single and multiple calling, so multiple
calling presents no difficulty insofar as getting the deviation from mean
chance expectation is concerned.
To illustrate
the calculations of the variance in a hypothetical set of data, let us assume
that 10 subjects called the same set of 25 ESP targets. For targets of either
type, (1) or (2), it is necessary that the distribution of selected symbols on
each set of calls for a single target be obtained. These may conveniently be
arranged as shown in Table I.
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Table 1 ASSUMED DATA FROM TEN SUBJECTS CALLING ONE SET OF 25 TARGETS Distribution
of Calls |
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|
Target Number |
Waves |
Circle |
Star |
Square |
Cross |
Total Calls for Each Target |
|
1 |
2 |
1 |
3 |
1 |
3 |
10 |
|
2 |
1 |
2 |
4 |
0 |
3 |
10 |
|
3 |
3 |
2 |
3 |
1 |
1 |
10 |
|
4 |
1 |
1 |
3 |
2 |
3 |
10 |
|
5 |
1 |
3 |
2 |
2 |
2 |
10 |
|
6 |
1 |
1 |
5 |
2 |
1 |
10 |
|
7 |
3 |
2 |
3 |
2 |
0 |
10 |
|
8 |
2 |
1 |
2 |
2 |
3 |
10 |
|
9 |
2 |
3 |
3 |
1 |
1 |
10 |
|
10 |
1 |
1 |
2 |
3 |
3 |
10 |
|
11 |
3 |
0 |
4 |
2 |
1 |
10 |
|
12 |
2 |
2 |
3 |
1 |
2 |
10 |
|
13 |
0 |
3 |
2 |
3 |
2 |
10 |
|
14 |
1 |
2 |
5 |
0 |
2 |
10 |
|
15 |
2 |
3 |
2 |
2 |
1 |
10 |
|
16 |
1 |
3 |
4 |
1 |
1 |
10 |
|
17 |
2 |
2 |
3 |
2 |
1 |
10 |
|
18 |
2 |
2 |
2 |
2 |
2 |
10 |
|
19 |
0 |
2 |
3 |
3 |
2 |
10 |
|
20 |
0 |
3 |
4 |
2 |
1 |
10 |
|
21 |
1 |
3 |
5 |
1 |
0 |
10 |
|
22 |
2 |
2 |
3 |
1 |
2 |
10 |
|
23 |
3 |
1 |
2 |
3 |
1 |
10 |
|
24 |
1 |
3 |
3 |
3 |
0 |
10 |
|
25 |
2 |
3 |
4 |
0 |
1 |
10 |
|
Total Calls for each symbol |
39 |
51 |
79 |
42 |
39 |
250-Total calls |
(1) Variance for
random targets:
If the ten
subjects of the hypothetical group were calling randomly selected targets, the
results of each trial would be statistically independent of the others, and the
variance of the number of correct calls on each target is given by the
expression:
(Probability of each target) x (Sum of
squares of calls for each symbol) - (Square of expected number of correct
calls).
For the first
target in Table 1, this is:
(1/5) x (22 + 12 +32
+ 12 +32) - (22) = 4 /5.
The variance for
each of the other targets is computed similarly, and the total variance for the
entire set of 250 calls is obtained by summing the variances for the 25
targets. If the calls of Table I were made against random targets, this total
variance is found to be 30.4
The standard
deviation is the square root of this value, i.e. 5.51. This may be compared
with the standard deviation of 
= 6.32 for the situation in which each of 10
subjects makes 25 calls for a different randomly selected
order of 25 targets.
(2) Variance for
the balanced or closed ESP pack :
If the targets
were provided by a randomly selected permutation of the standard ESP pack, the
variance for data involving multiple calling must be obtained for the entire
set of 25 targets as the unit, and the computations
are more involved than for data based upon random targets. The quantities
required in the calculations are:
(a) The number of ESP symbols =
5
(b) The number of targets =
25
(c) The sum of squares of the call frequencies in each of the 250 calls of Table I
22 + 12 +32
+ 12 +32 +12 +…+42 +02 +12
= 652
(d) The sum of the squares of the total calls
of each symbol
392 +512 +792
+422 + 392 =
13,648
(e) The sum of the squares of the total calls
for each target
(25) (102) =
2,500
(f) The square of the total number of
calls
2502 =
62,500
Then, for the
closed pack:
Variance 
This may be
compared with the variance of 41.67 (S.D. = 6.46) which is the theoretical
value for the total score from 10 runs based upon a different set of 25 calls for each of 10 ESP closed packs.
The above
method for finding the variance of the balanced pack may be applied to multiple
calling data with runs of any length in which all targets are presented the
same number of times. It is not necessary for (b) to be the square of (a), as
happens to be the case in the
standard ESP pack of 25 cards.
The
Greville method applied to the assumed set of calls shown in Table I gives a
lower variance for both (1) and (2) than the method appropriate for single
calls. In actual practice, it is generally found to give a higher variance. In
all events, the only safe procedure is to use the correct method. The variance
for both (1) and (2) may be obtained regardless of whether the same number of
calls are made on each trial. It will be seen that the computation for the
closed pack is simplified some what if every subject makes a call on every
target, as then (be) = f and these two quantities cancel out in the formula
given above.
For (I), it is not necessary to deal with the data in standard runs, but the series may be of any desired length.
For (2), the variance must be computed for
each set of calls made upon a random permutation of the closed pack, and the
separate variances of the sets of calls for each pack of 25 targets are added
to get the total variance for the data as a whole.
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Parapsychology
Laboratory Duke University Durham, North Carolina |
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© Copyright Bernard AURIOL (email : )
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