Journal of Experimental Psychology

Volume 21, 1937

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AN EXAMINATION OF THE COMPUTING ABILITY
OF MR. SALO FINKELSTEIN
A. ADMINISTRATION, DESCRIPTION OF THE TESTS, AND
GENERAL PSYCHOLOGICAL CONCLUSIONS
BY JAMES D. WEINLAND
B. STATISTICAL ANALYSIS OF THE CORRELATION BETWEEN
NUMBER OF SEPARATE ACTS OF ATTENTION IN TESTS,
WITH TIMS REQUIRED, AS WELL AS WITH
ACCURACY OF PERFORMANCE
BY W. S. SCHLAUCH

I. CHARACTER AND BACKGROUND OF THE SUBJECT

Salo Finkelstein, Jewish, 35 years old, born in Lodz, Russia, now Poland, and carrying references proclaiming him a 'lightning calculator' to be compared with Dr. Ruckle, Inaudi or Diamondi, presented himself at the laboratory of New York University, School of Commerce, and offered to subject his calculating ability to examination and test.
He gave the following information: As a boy in school he did his number work more quickly than the average student, and liked arithmetic better than history or geography, but he had no premonition of the calculating ability which he was later to discover in himself. When he was 23 years old, and out of school, a friend of his claimed that he could multiply mentally two numbers of three digits each. F. tried it and found he could do even better. He easily multiplied in his head two numbers of six digits each! F. analyzed the process and found that to multiply numbers by memory, it is not necessary to be a good calculator, but it is necessary to have a good visual imagery to keep the numbers in mind. He tried to repeat both acoustically and visually presented numbers. He did it well and began demonstrating for the public. He lost interest however and gave up the practice till his 27th year when he read a newspaper account of a man with a good

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memory for numbers. He recalled his own success of the previous years and tried again. This time he gave demonstrations to which medical men, teachers, scientists and the newspapers were invited. He tried to do things never before attempted and did them without mistake. Soon after this he was employed in the State Statistical office of the Polish Government. He frequently was assigned to the treasury department on special problems where especially quick and exact work was necessary. In calculating budgets, sums must be combined in various ways, and the results known quickly. This he did for the treasury. He worked for the government for 11 years and in that time not a single error was found in any of his calculations. The explanation of this is that he always checked his work, doing the problem In sosae other way than at first. It is his custom still to check each calculation he makes and in his public performances he usually follows his statement of the answer with the words, "check ... no mistake." If any mistake is found after he gives the signal, F. has made the standing offer that he will pay $100 to have it pointed out.
In 1928, F. was invited by Professor Hans Henning in Danzig for a test. As Prof. Henning had previously tested Ferroll and Ruckle he could make a comparison. He found F. a superior calculator but was most interested in the associations he found with numbers. F. had an aesthetic feeling an regard to numbers. He felt in them a beauty comparable to music or art. In 1930, at the age of 33 F. was invited to the international convention of insurance mathematicians in Stockholm. This group confirmed the previous judgments that his abilities were marvelous. In 1931 F. gave up his position with the government to go around the world to demonstrate his abilities and submit to scientific tests. He was tested by Bela Sandor in the Institute of Industrial Psychotechnics in the Technical Highschool in Berlin. This investigation is reported in Character, 1, 1932, page 47. Sandor found his memory for letters average, his memory for drawing below average. In the United States F. demonstrated in a number of universities. For a period covering

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several months F. came frequently to the School of Commerce, New York University, and submitted to the tests in calculation presented here. He did not wish to take an intelligence test. His calculating abilities speak for themselves.

II. THE SUBJECT'S ATTITUDE TOWARD HIS OWN ABILITY

F. can analyze and give some insight into his own mental processes. He has studied psychology, especially that relating to memory, and knows of the performance of the preceding lightning calculators. But with all his ability to manipulate numbers, F. is not a mathematician. By using his special abilities he is able to make calculations he believes of value to mathematicians and also to many others. For instance, concerning mathematical processes, he is able to reduce numbers to the sum of three or four squares very rapidly. Thus he reduced 9,413 to the sum of 95² plus 12² plus 12² plus 10² in ten seconds. No principle or law explaining how this can be done is known.
So far as we can observe, F. works merely by trial and error, but works so rapidly that it would be possible to gather a great amount of data concerning his methods in a short time. General principles might possibly be derived if we had more detailed and elaborate data on his methodology.

III. THE TESTS AND THE DATA THEY YIELDED

Various devices were used in the calculating tests. A pendulum chronoscope was connected with a balloptican, in such a way that time was measured as long as the figures were projected on the screen. F. threw the light on and snapped a switch in his hand, turning it off, as soon as he felt that he could answer the problem. The whipple statistoscope was used for some exposures. The calculating tests requiring more than four seconds were measured with a stop watch. A paper with a problem written on it was presented to F., or while his back was turned the problem was written on the blackboard. At the signal, beginning the time measurements, F. turned, made a line through the numbers and wrote down the result. Drawing the line seemed to help him, as though

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"he was prodded with a stick." When he was trying to work very fast he would draw the line with more emphasis. In difficult experiments he has noticed the strain on his body. He says he has had a pulse of 140 in some performances. "I feel not so bad today because the artist in my personality defends me from it, and seeing my success counterbalances the weakness I experience during the performance." He reported that he feels the strain most when obliged to make experiments outside the field of numbers where his ability is 'above average but not phenomenal.' He showed a wanning up curve with increasing concentration on each test day. He would worfk om his calculation for half a day at a time. He claimed he did not get tired. He said there was no work to it ... but he would pace back and forth under high nervous tension. He showed slight distractibility and little fatigue. He believes concentration to be the dominating factor. He eliminates other things that might influence. In more difficult calculations, memory is the most important thing. It is important in recalling at the proper time the right associations and the right method of work to deal with the vast quantity of material in the mind. F. states that he never practices except in giving demonstrations. He claims that he discovered his ability and did not develop it. He has an active Intense preference for complexes rather than for individual numbers. It worried him considerably if he made several mistakes in succession. He then would stop for the day. He explained that to work so very rapidly he had to have confidence. Anything that disturbed his confidence made it Impossible to work. He claimed that he did not have eidetic imagery and stated In evidence of this that no matter how digits were presented to him, no sooner was the exposure ended than he saw the digits in his own handwriting. He is not dependent on mood or health. He can notice changes in these respects but others can't tell the difference. He believes his attention and concentration are always above the average; sometimes extraordinary. He could add or multiply much more rapidly than he could divide, and preferred these processes. Only once did he do a problem in division at New York University,

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dividing mentally 67,916,288 by 8,558 and giving the correct answer in three minutes. He knows 300 decimals of Pi. He knows logarithms from 1-100 to 7 decimals. He knows logarithms from 100-150 to 5 decimals. This helps him in his calculations.
He remembers many hundreds of dates, but only when the matter is specially interesting or when the number is 'nice.' 0 for instance is not 'nice.' The nicest numbers are combinations that happen to please. A great many tests were given F. at N. Y. U. The results of a few of these tests are shown in Tables I-III. TABLE I TIME IN SECONDS OF HORIZONTAL AND VERTICAL ADDITION

Number of
Columns Number in
Column Total No.
Digits Addition Time in Seconds

Horizontal Vertical

1
2
3
4
5
6
10 60
30
20
15
12
10
6 60
60
60
60
60
60
60 15.4
15.7
19.7
14.0
17.3
15.1
17.3 13.8
15.9
16.5
19.5
15.7
17.6
16.9

Methods: F. reports that simple numbers mean nothing to him. He likes them only when they are in complex form. His mental processes in adding the digits 97438759874 he gave as follows: 9 + 7 + 4 + (3 + 7) + (8 + 5) + (7 + 4) + (9 + 8) = 71. In his mental calculations F. does not use mnemonic methods. He does not reduce the numbers to factors. When he multiplies in memory or on the blackboard, he uses well known methods from commercial arithmetic. He made use of the distributive law, multiplying 58 x 43 in seven seconds as follows: 50 = ½ of 100. ½ x 43 = 21.5, .^.. 50 x 43 = 2,150. 8 x 43 = 344. 2,150 + 344 = 2,494. By the factor method he multiplied 44 x 88, writing the answer at once. 44² = 1,936. 88 = 2 x 44. 1,936 x 2 = 3,872. By single factor multiplication he wrote the product of 77 x 33 immediately. 3 x 77 + 231 (x 10) + 231 = 2,541.

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TABLE II REDUCING TO SUM OF THREE SQUARES

Number Result
Reported** Time (Seconds)

129
99
74
67
53
45
37
33
29
27
25
21
13
10
876
645
944
895
488
111
832
472
907
391
395
*555
101
186
487
921
456
876
579
753
749
747
745
743
741
425
427
133
131 11-2-2
9-3-3
(8-3-1)-(7-4-3)
7-3-3
6-4-1
5-4-2
(Cannot be done)
(4-4-1)-(5-3-2)
4-3-2
(5-1-1)-(3-3-3)
(Cannot be done)
4-2-1
(Cannot be done)
(Cannot be done)
(26-10-10)-(26-14-2)
23-10-4
28-12-4
(Cannot be done)
18-10-8
(Cannot be done)
(Cannot be done)
20-6-6
27-13-3
(Cannot be done)
15-13-1
23-5-1
9-4-2
13-4-1
(Cannot be done)
29-8-4
(Cannot be done)
26-14-2
23-7-1
20-17-8
27-4-2
23-13-9
22-15-6
(Cannot be done)
23-14-4
20-14-3
15-11-9
9-6-4
11-3-1 2.0
1½
4½
(?)
5.0
7.0
16.0
3.0
3½
1½
2.0
(?)
(?)
(?)
7.0
4.0
5.0
1 minute
6.0
(?)
2 min. 10 sec.
1½
3.0
(?)
3.0

1.0
7.0
(?)
12.0
(?)
7.0
8½
6.0
4.0
8¾
1 min. 56 sec.
1 min.
6.0
1.0
1.0
(?)
1.0

*Note: Dissatisfied with first large number, tried to find numbers more nearly equal in size; couldn't.
** This column means, e.g., 129 = 11² + 2² + 2²; 488 = 18² + 10² + 8².

He used the cross product method a great deal. An example of the mental operations necessary to multiply a four place number by a four place number are as follows. In the mental

- 388 -
TABLE III REDUCING TO SUM OF FOUR SQUARES

Number Result
Reported* Time (Seconds)

863
1486
9625
1100
4444
6777
366
8123
4567
7889
753
6328
9417 27-11-3-2
38-8-4-1
95-22-10-4
24-22-6-2
62-22-10-4
76-31-6-4
16-10-3-1
81-39-5-4
61-29-2-1
81-36-4-4
24-13-2-2
32-20-2-70
95-12-12-10 60
12
6
25
15
15
3.5
50
25
30
26.5
10
10
* This column means, e.g., 863 = 27² + 11² + 3² + 2²; 9413 = 95² + 12² + 12² + 10².
multiplication all of these operations must be held in mind till the final answer is given.

By the use of this method F. multiplied 6,943 x 7,859 and gave the correct answer in 17.5 seconds.
F. stated that there is no known method of reducing numbers to the sum of squares. It must be done by trial and error. All numbers above a certain number can be reduced to the sum of four squares, but certain ones can not be reduced to the sum of three squares. Mental operations reported by F. in reducing certain numbers are as follows: Problem: reduce 6,328 to the sum of four squares. Thought that 71² = 5,041. Thought of subtracting it; didn't like it, so didn't. Thought 72². Doesn't know it. 70² = 4,900 subtracted from 6,328 = 1,428. Has it. 1,428 into 3 squares equals 32² + 20² + 2², .^.. 6,328 = 70² + 32² + 20² + 2². Problem: reduce 9,413 to 4 squares. Took 91², it is 8,281. Subtracted from

- 389 -
9,413, result is 1,132. Subtracted 30², result is 232. 232 into two; 144 = 12²; not good; 32² = 1,024; subtract from 1,132 = 108. Rejects this also since he can't reduce this to two squares. Threw 91 away; took 95² = 9,025. Subtracted, 388. Thought of 324 = 18²; not good. Took 16² = 256. 388 - 256 = 132; not good. 388 = 2 x 144 + 100. Answer 95² + 12² + 12² + 10².
Time: Ten seconds for each of the above problems, both done mentally.

IV. GENERAL CONCLUSIONS DRAWN FROM THE TESTING PROCESS

(A) Though F. calculates much more rapidly than the average person, his thinking processes are responsive to the same mental laws. His own idea that practice has no effect does not seem to be substantiated. This is evident from his work in division. He seldom demonstrates in division, has little practice in it, finds it somewhat difficult, and is comparatively slow. F.'s learning of numerical facts and associations has been very rapid, but it is nevertheless learning of the usual kind. His 'lightning calculations' all require time for their performance. Problems solved 'immediately' were not solved but recognized. F. had previously worked them and remembered the result. The fact that the solution-time is so often very short is an indication of skill and native quickness, not of any unnatural powers.
(B) Concentration, memory, and confidence are among the most necessary mental qualities for this kind of work.
(C) The fact that F. added horizontally, with little practice, as rapidly as vertically after much practice, is significant. He required much less practice in horizontal addition. The speed may have indicated transfer of skill from the eye movements of reading, or it may be due to the wider span of horizontal than of vertical perception. With equal practice horizontal addition should be faster than vertical addition. The results indicate that, disturbing conventional accounting and bookkeeping systems aside, the speed of addition for people in general might be increased by teaching horizontal rather than vertical addition.

- 390 -
B.. STATISTICAL ANALYSIS OF THE DATA
BY
W. S. SCHLAUCH

The general psychological conclusion that even in the case of a computing prodigy, time is required to complete an addition or a multiplication naturally raises the questions, "How much time is consumed?" and "Does the time consumed vary in any regular way with measurable elements of the computation problems solved?" The tests show that in the case of addition, the time needed to add depends on the number of digits added, and is not appreciably affected by the number of columns into which the digits are arranged. In multiplying numbers of varying numbers of digits, the evidence indicates that the time varies not with the number of digits, but with the separate acts of attention involved in the process, which rapidly increases as the number of digits is increased.
The last statement is true also of addition, as the number of acts of attention here varies with the number of digits. So that in computation, whether addition or multiplication, the correlation seems to be between number of separate acts of attention and time taken, and also between number of separate acts of attention and accuracy of the result.
It is important to know, not only that the time taken increases as the number of acts of attention involved increases, but also the law of increase or variation. Is the increase of time needed in proportion to the number of acts of attention, or does it vary as the square of the number oг separate acts of attention? These questions can be answered, provided the data are adequate, by a statistical analysis. We proceed, therefore, to a study of the correlation problems involved in these experimental data.

V. CORRELATION OF TIME CONSUMED IN ADDITION, WITH NUMBER OF DIGITS TO BE ADDED

In adding a column of digits there is a separate act of attention for every digit added to the sum of the digits pre-

- 391 -
viously united. (Thus the span of attention is widened approximately in proportion to the number of digits to be united.) The number of columns of digits also enters into the problems slightly as, for example, 24 digits in one column involve 24 acts of summation of each succeeding digit with the units digit of the previous sum and the corresponding 'carry.' If the 24 digits are in two columns we have the same number of 'unite' and 'carry' acts, but there is a break in registering the units digit of the sum of the first column before continuing with the second column. There is also the shift after the addition in the second column of again considering the digits of this result as units of the next higher order. There would thus be 25 acts of conscious manipulation involving attention. However, this element disturbs our results only slightly, as may be seen from the graph showing the relation of time in seconds to accomplish horizontal and vertical addition to the number of digits added.
The moving average of the time for vertical addition is also shown on the graph. It follows a straight line trend so closely that we are justified in finding linear correlation of these elements. FIG. 1. Graph of time to add, in seconds.

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Correlation of Time to Add with Number of Digits Added in Vertical Addition
Using the data of Table IV we may derive the equadon of the regression line of time to add on number of digits added, by using the individual items. For this purpose we set up a table in which the items in the column 'Number of Digits' are called X, those in the column headed 'Observed Time' are TABLE IV FINKELSTEIN TESTS Time in Seconds for Vertical and Horizontal Addition

No of
Digits Vertical Adding
Time Horizontal Adding
Time No of
Digits Vertical Adding
Time Horizontal Adding
Time

Ob-
served Moving
Average Ob-
served Moving
Average Ob-
served Moving
Average Ob-
served Moving
Average

27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63 4.8
5.4
6.5
6.7
7.5
6.2
7.0
7.3
7.5
7.6
7.6
8.7
8.7
8.1
8.6
10.3
11.6
11.2
11.2
10.7
11.2
11.3
12.4
11.8
12.8
10.5
12.6
14.0
12.4
13.4
13.0
14.2
12.4
13.8
14.3
16.4
14.4

6.2
6.5
6.8
6.9
7.1
7.1
7.4
7.7
8.0
8.1
8.3
8.9
9.5
9.9
10.6
11.0
11.2
11.1
11.1
11.3
11.7
11.5
11.8
12.1
12.2
12.4
12.9
13.2
12.9
13.1
13.3
14.0
14.0
14.6
15.1 4.3
6.5
6.5
6.7
6.1
6.2
6.0
6.0
6.7
7.7
7.5
9.3
8.5
7.2
8.5
8.5
12.5
10.8
10.0
11.7
11.4
11.2
14.9
12.0
10.8
12.8
11.5
12.5
13.2
15.5
12.8
15.5
13.0
15.4
18.5
15.8
15.2

6.0
6.4
6.3
6.2
6.2
6.5
6.8
7.4
7.9
8.0
8.2
8.4
9.1
9.5
10.1
10.7
11.3
11.0
11.9
12.3
12.1
12.4
12.4
11.9
12.2
13.1
13.1
13.9
14.0
14.5
15.1
15.7
15.6
16.4
16.4 64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100 15.0
16.4
20.2
17.3
20.2
19.0
16.6
18.0
17.4
20.3
21.0
20.5
21.0
24.3
20.4
20.2
22.8
21.9
28.3
20.6
24.9
21.2
22.8
27.9
21.1
23.5
21.0
21.8
24.2
25.5
25.5
27.1
26.9
25.6
30.6
26.5
26.7
16.3
16.4
17.8
18.6
18.7
18.2
18.2
18.3
18.7
19.4
20.0
21.4
21.4
21.3
21.7
21.9
22.7
22.7
23.7
23.4
23.5
23.5
23.6
23.3
23.3
23.1
22.3
23.2
23.6
24.8
25.8
26.1
27.1
27.3
27.3

17.0
15.6
16.8
15.2
18.5
19.5
17.4
19.8
17.5
18.8
20.8
19.1
17.0
22.2
22.8
18.4
19.0
20.4
28.0
22.4
24.2
22.2
22.3
22.4
23.8
23.6
22.2
25.0
24.2
24.6
25.8
25.4
26.2
27.0
26.0
26.8
25.6 16.1
16.0
16.6
17.1
17.5
18.1
18.5
18.6
18.8
19.2
18.6
19.6
20.4
19.9
19.9
20.5
21.7
21.6
22.8
23.4
23.8
22.7
23.0
22.8
22.8
23.4
23.7
23.9
24.4
25.0
25.3
25.8
26.1
26.3
26.3

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called Y. The column headings of the table are then: X, Y, X²XY, and Y². The totals of these columns are respectively: 4665, 1201.0, 330 993, 86 939.5, and 23 112.92.
The table itself is omitted to save space. From these totals we find:

M_x = 4665
73 = 63.904, M_y = 1201
73 = 16.452.

For the equation of the regression line, Y' = a + bX,

b = SXY - nMxMy
SX² - nMx² = 86939.5 - 73 x 63.904 x 16.452
330993 - 73 x 63.904² = 0.31.

a = My - bMx = 16.452 - 0.3099 x 63.904 = - 3.35.
The regression equation is: Y' = - 3.35 + 0.31X, where Y' is the most probable time in seconds to add, and X is the number of digits added. Thus, if we wish to find the most probable time necessary to add 80 digits, Y' = - 3.35 + 0.31 x 80 = 21.45 seconds.
The experiments showed actual time taken was 22.8 seconds. The coefficient of correlation is given by

r = SXY -nMxMy
√[SX² - nMx²][SY² - nMy²] = 10191.0
√32881.4 x 3354.1 = 0.972.

If we adjust for number of samples, calling the adjusted coefficient r, we find r = 0.968.
The high coefficient of correlation indicates that a very close estimate of the time it will take this prodigy to add a given number of digits can be made by using the regression equation. To measure how accurate the estimate is, we pro-

- 394 -
ceed to find the standard error of the estimate. For this purpose, a column of values of Y', corresponding to each given X of the experiment, was calculated; then a column of values of Y - Y' (or Z), and a column of values of Z².
The Z² column total is 190.71.
The unadjusted standard error is

S = Z
n - 1 = 190.7
72 = 2.64875,

from which S = 1.62 sec.
The adjusted standard error is S = 1.64 sec.
This means that there is about a 2 to 1 chance that the actual time taken in such an addition problem will not differ from the estimated time by more than 1.6 seconds. FIG. 2. Correlation of vertical addition time with number of digits added.

- 395 -

These facts are illustrated in the following graph, Fig. 2. RR is the regression line, and the dotted lines SS are the limits of the standard error of the estimate.
The rather wide fluctuation of the time required for the addition when the number of digits exceeded 80 makes S wider than would be the case, if the standard error had been calculated from the specimens whose number of digits ranged from 27 to 76.
The conclusion seems safe that the time needed for executing the mental processes involved in adding digits is a linear function of the number of digits added.
It also appears that for this individual at least, residual factors such as a feeling for certain digits, or other like elements, play a small part in the process. The memory factor, as well, requires no more time as the number of digits increases.

VI. STATISTICAL INTERPRETATION OF MULTIPLICATION TESTS

In these tests, examples were given, covering the multiplication of two digits by two digits, three digits by three digits, four by four, and five by five. The number of separate acts of attention called for in these four classes of examples were respectively six, twelve, twenty-one, and thirty-two. The data are thus somewhat scanty from which to draw valid conclusions as to the regression line. If three digits by two digits, three by four, etc., had been included, we should have some basis for deciding whether the regression line is linear or follows a parabola of the type y = ax² + bx + c.
Since the multiplying of one number by another involves not only a certain number of acts of attention which are easily accomplished because they have been previously routinized, but involves as well (and simultaneously) the mental arrange-

- 396 -
ment of partial results according to order of digits, as units, tens, etc. and then their summation as additional separate acts of attention, it may well be that the regression is actually a curved line. Certainly the second degree parabola would fit the data with a smaller standard error than the straight line does. Additional tests must be made to determine the real nature of this regression curve. The straight line regression affords a good estimate of the time needed by Mr. Finkelstein for mentally multiplying numbers up to five digits by five digits.
If we average the time needed to complete each of the four types of example used in the multiplication test, and find the percent of each type solved correctly we have: TABLE V AVERAGE TIME AND ACCURACY IN MULTIPLICATION

Type of Example

1 Range of Time
to Complete
(Seconds)
2
No. of
Examples
3 Average Time
for Type
(Seconds)
4
Percent
Correct
5

2 by 2 (6 acts) 2.0-14.0 30 4.03 93.3

3 by 3 (12 acts) 5.5- 5.6 6 5.7 83.3

4 by 4 (21 acts) 12.0-18.5 10 13.5 60.0

5 by 5 (32 acts) 19.2-28.6 9 25.8 44.4

Plotting columns 1 and 4 shows positive correlation, while columns 1 and 5 show decided negative correlation. The linearity of the first is in doubt, but that of the second is not doubtful, so far as these four types are concerned.
Since the number of items is not great we use the separate items in finding the coefficient of correlation and the equation of the regression line. This, of course, gives a more accurate result than if a double entry table had been used. The details of the calculation can be understood by inspecting the following table and the formulae following it.

- 397 -
TABLE VI

No.
Acts of
Atten-
tion
X Time
in
Sec-
onds
Y F FX FY FX² FY² FXY Y' Z=Y-Y' FZ²

6
6
6
6
6
6
6
6
6
6
6

12
12
12

21
21
21
21
21
21

32
32
32
32
32 2.0
7.0
4.0
9.0
14.0
1.0
3.5
4.7
5.5
2.8
3.1

5.5
5.5
5.6

18.5
12.0
17.5
12.3
13.0
12.1

19.2
26.3
28.6
25.8
26.0
1
1
1
1
1
4
3
4
4
5
5

1
2
3

1
1
1
2
2
3

1
2
2
2
2
6
6
6
6
6
24
18
24
24
30
30

12
24
36

21
21
21
41
42
63

32
64
64
64
64 2.0
7.0
4.0
9.0
14.0
4.0
10.5
18.8
22.0
14.0
15.5

5.5
11.0
16.8

18.5
12.0
17.5
24.6
26.0
36.3

19.2
52.6
57.2
51.6
52.0 36
36
36
36
36
144
108
144
144
180
180

144
288
432

441
441
441
882
882
1323

1024
2048
2048
2048
2048
4.00
49.00
16.00
81.00
196.00
4.00
36.75
88.36
121.00
39.20
48.05

30.25
60.50
94.08

342.25
144.00
306.25
302.58
338.00
439.23

368.64
1383.38
1635.92
1331.28
1352.00

12.0
42.0
24.0
54.0
84.0
24.0
63.0
112.8
132.0
84.0
93.0

66.0
132.0
201.6

388.5
252.0
367.5
516.6
546.0
762.3

614.4
1683.2
1830.4
1651.2
1664.0
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.4

8.2
8.2
8.2

15.4
15.4
15.4
15.4
15.4
15.4

24.2
24.2
24.2
24.2
24.2 - 1.4
+ 3.6
+ 0.6
+ 5.6
-10.6
- 2.4
+ 0.1
+ 1.3
+ 2.1
- 0.6
- 0.3

- 2.7
- 2.7
- 2.6

+ 3.1
- 3.4
+ 2.1
- 3.1
- 1.4
- 3.3

- 5.0
+ 2.1
+ 4.4
+ 1.6
+ 1.8
1.96
12.96
0.36
31.36
112.36
23.04
0.03
6.76
17.64
1.80
0.45

7.29
14.58
20.28

9.61
11.56
4.41
19.22
11.52
32.67

25.00
8.82
38.72
5.12
6.48

55 750 521.6 15570 8811.2 11400.5 424.00

M_x = 750
55 = 13.636, M_y = 521.6
55 = 9.484.

For Regression line Y' = a + bX,

b = SXY - nMxMy
SX² - nMx² = 11400.5 - 55 x 13.636 x 9.484
15570 - 55 x 13.636² = 0.8024.

a = My - bMx = 9.484 - 0.8024 x 13.636 = - 1.456.
The Regression Equation is: Y' = - 1.458 + 0.802X,
where X = No. acts of attention involved in problem,

Y'= No. seconds required to solve on the average.

- 398 -

For the Coefficient of Correlation:

r = SXY - nMxMy
√[S(x²) - n(Mx)²][S(Y²) - n(My)²] ,

.^..r = 4287.69
5343.27 x 4947.04 = 4287.69
5138.93 = 0.8344.

If we adjust r for the number of samples, r being the adjusted coefficient, as r is the unadjusted coefficient, we have: r = 0.831.
We may thus estimate the time in seconds it will take this prodigy to find the result in multiplying, up to 5 digits by 5 digits by the equation Y' = - 1.458 + 0.802X, and most of the variation of time required is accounted for by the number of separate acts of attention involved in the multiplication of the separate digits and the addition of the partial results.
To find the standard error of the estimate made from this regression line we find Y' for X = 6, 12, etc., and find the residuals of the separate trials.
If X = 6,
X = 12,
X = 21,
X = 32, Y' = - 1.458 +
Y' = - 1.458 +
Y' = - 1.458 +
Y' = - 1.458 + 4.812 =
9.624 =
16.842 =
25.664 = 3.354.
8.166.
15.384.
24.206.

These values, given by the regression equation, fall between the actual averages. We use these values in columns 9, 10 and 11.
The unadjusted Standard Error of the Estimate (made by our regression line), is

Sxy² = Z²
n - 1 = 424.00
54 = 7.8518.
.^.. Syx = 2.80 sec. or 2.8 prac.
For the adjusted Standard Error: Syx = 2.83 or 2.8 practically.

- 399 -

This means that if we estimate the time it will take Mr. Finkelstein to finish a problem in multiplication of the above type from the number of acts of attention involved in the problem, and repeat this a considerable number of times, approximately 68 percent of the actual times would not differ from their corresponding estimated times by more than 2.8 seconds.
On the accompanying graph, the Regression line is shown, and a band on either side, 2.8 seconds wide, limited by the dotted lines, shows the region within which 68 percent of the estimates will fall.
The separate black dots (.) on the diagram show the actual time records of the test. The open circles (o) show the average of the trials in which six acts of attention, 12 such acts, etc., were involved. FIG. 3. Correlation of time required with number of separate acts of attention in multiplication.

- 400 -
Probable Curved Regression Line for Time Required in Multiplying
If we use the method of least squares to fit a second degree parabola to the average data:
No. Acts of Attention (X)
6
12
21
32 Time to Complete (Y)
4.03
5.70
13.50
25.80
we find that the equation is: Y' = 0.0212X² + 0.0473X + 2.6994.
The calculated values for Y (Y') for X = 6, 12, 21, 32 respectively are: 3.75, 6.32, 13.04, 25.92.
This, of course, is a better fit than the straight line, and would greatly reduce the standard error of the estimate. If the data for 9, 12, etc. acts of attention fell near this parabola we should be reasonably sure that for multiplication, the time of performance increased faster than shown by the linear regression, and tended to vary as the square of the number of acts of attention.
This parabolic regression line is indicated on the graph, labeled PP, while the linear regression is labeled RR.

VII. REGRESSION OF ACCURACY ON NUMBER OF SEPARATE ACTS OF ATTENTION IN MULTIPLICATION

It may be seen from Table V that 93.3 percent of the problems involving six separate acts of attention were solved correctly by Mr. Finkelstein; 83.3 percent of those involving twelve acts of attention were solved correctly, 60 percent of those involving 21 acts, and 44.4 percent of those involving 32 acts of attention.
If these data are plotted, as shown on the graph following, it is evident that there is negative correlation between percent accurate and number of acts of attention involved and that the regression is linear. By using the equations of condition aSx² + bSx = Sxy,
aSx + bn = Sy,

- 401 -
we find the coefficients a and b of the regression line y = ax + b.
Substituting the values of Sx², Sx, Sxy etc. (where x is the number of acts of attention, and y the percent accurate), 1645a + 71b = 3886.5,
71a + 4b = 269.9. Solving, a = -2.35
b = 109.19. FIG. 4. Regression of accuracy on no. of acts of attention. y = - 2.35x + 109.19.
The equation, showing how accuracy declines, on the average as the number of separate steps of attention increases is thus y = - 2.35x + 109.19.
To test the accuracy of fit of this line, we find calculated average values of y, to compare with the observed average values.

- 402 -

If
x = 6,
x = 12,
x = 21,
x = 32, (Calc. y)
y = 95.1
y = 81.0
y = 59.8
y = 34.0 (Obs. y)
93.3
83.3
60.0
33.3
(Devia.)
+ 1.8
- 2.3
- 0.2
+ 0.7

It may be seen that the sum of the deviations of the observed average accuracy percents from the calculated values is zero.
If our regression line equation gives the correct formulation of the relation involved, we should be able to calculate within very narrow limits of error the percent of problems this prodigy would solve correctly for any combination of digits. Thus, if he tried multiplying four digits by three digits, as 7,345 x 639, and tried such problems a number of times we should expect that y (percent accurate) would be, since x = 17, y = - 2.35 x 17 + 109.19 = 69.2.
That is, we should expect him to have about 69 percent of such problems correctly solved.
General Conclusions as to Multiplication
1. The time consumed in multiplying one number by another mentally increases about 0.8 second for each additional act of attention involved, for problems of two digits by two digits to those of five digits by five digits. (On the assumption that the regression is linear.)
2. As the number of acts of attention (or mental manipulations) is increased, the accuracy decreases rapidly. The decrease in percent of accuracy is 2.35 percent for each additional act of attention involved in the Example, on the average. (Manuscript received March 29, 1937)