College and Research Libraries


A Statistical Method for 
Circulation Analysis 

WHAT is the best way to distribute money 
allocated for the purchase of books? 
Should more money be spent on addi-
tional copies of the very popular works, 
or should a greater variety of books be 
bought? 

Some techniques borrowed from the in-
dustrial inventory control specialists will 
help answer these and other questions 
posed · by progressive library manage-
ments. One particularly applicable prob-
lem of industrial concern is the predic-
tion of stock levels of returnable items. 
For example, a tool room lends many 
thousands of different types of tools to 
the mechanics it serves. Assurance of an 
adequate supply of each type of tool is 
vital. Furthermore, a limit on the amount 
of money to procure these tools demands 
a rational and exacting method of decid-
ing what and how much to buy. 

This problem is similar to those librar-
ies where high demand calls for multi-
ple copies of many books. With mechan-
ical and electronic data processing, 1 the 
analysis below has been successfully ap-
plied to the tool crib problem above, and 
can be used with little modification for 
the · corresponding library management 
problem. 

The high turnover pattern of a partic-
ular book2 can be shown graphically as 
in Figure 1. Each horizontal line repre-

1 R. W. Gibson and G. E. Randall, "Circulation 
Control by Computer," Special Libraries, LIV (July-
August 1963), 333-38. 

William L. Leffier, "And Now, Computer Tool Con-
trol," Mill and Factory, LXXIII (September 1963), 
92. 

2 In this context, a "book" refers to a particular 
title, author, edition, etc., of which a library has 
multiple copies. 

BY WILLIAM L. LEFFLER 

Mr. Leffler is with the Shell Oil. Company, 
50 West 50th Sf,reet, New York 20. 

sents the borrowing and returning of a 
copy of the book and the length of time 
it was kept out. 

During the forty days of observation, 
fifteen loans were made, fifteen returns 
were made. Notice, however, that we do 
not need fifteen copies of our book. On 
the seventh day, when the sixth borrower 
requested a copy, the first copy was now 
available to be given to him. On the six-
teenth day, when the seventh borrower 
requested a copy, four more copies had 
been returned. Would it not be conveni-
ent to look at a chart like this and judge 
visually the maximum number of copies 
that were in use? It would be easy to de-
termine whether the stock of this book 
were being taxed to capacity, and if so, 
whether additional copies should be pur-
chased. 

Unfortunately, this simple analysis is 
an inordinately lengthy one, even for a 
high-speed computer. A more feasible 
approach is that taken in the tool crib 
problem-that of statistical prediction. 

Let us assume some parameters of our 
problem. Let T = number of days of ob-
servation; N =total number of loans; A1 = 
length of time a copy was kept out; A = 
average length of time all copies were 
kept out; S =(sum of all A1)/N. 

S = vSum of (Ai- A)2 
N 

= 2.58 

v 102.40 
15 

488 COLLEGE AND RESEARCH LIBRARIES 



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-----,1 Iteturn 

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Days 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 

FIG. 1.-The borrow-and-return pattern for a typical multiple copy book. 

N95% = Nav + 3s 
= 2.65 + 7.74 
= 9.39 

In our example graphed above in Fig-
ure 1, we find that T = forty days, N = 
fifteen loans, and Ai's are read off the 
graph, and A is the sum of all the A/s 
divided by the number of loans, or A = 
102/15 = 6.8. 

Empirical research has shown that the 
average number of books that are out on 
loan is calculated as follows: 

Nav = (N/T) X (A) 

In our example above, 

Nav = (15/40) X (6.8) 
= 2.65 

We must be careful, however, about 
interpreting this average. By its very defi-
nition an average implies that in half the 
cases the number will be less than the 

NOVEMBER 1964 

average, and in half the cases greater. 
Since we cannot be satisfied with know-
ing what the usage is in only half the 
cases, we must make statistical adjust-

TABLE 1 

CALCULATION OF THE STANDARD DEVIATION AND 
STOCK LEVEL OF 95 PER CENT CERTAINTY 

OBSERVATION At A (A t- A) (At-A)2 

1 6 6.8 -D.8 0.64 
2 4 6.8 - 2.8 7.84 
3 9 6.8 2.2 4.84 
4 2 6.8 -4.8 23.04 
5 11 6.8 4.2 17.64 
6 9 6.8 2.2 4.84 
7 5 6.8 - 1.8 3.24 
8 9 6.8 2.2 4.84 
9 7 6.8 .2 0.04 

10 8 6.8 1.2 1.44 
11 6 6.8 - 0.8 .64 
12 4 6.8 -2.8 7.84 
13 4 6.8 -2.8 7.84 
14 7 6.8 .2 0.04 
15 11 6.8 4.2 16.64 

105 Sumof(At-A)2 = 102.40 

489 



ments. Just as the average gives us the 
upper limit 50 per cent of the time, we 
can predict the level of loans that covers 
90 per cent, 9 5 per cent, or any other 
per cent. Since we are working with prob-
ability, we cannot give a level for 100 per 
cent certainty. So, as in the industrial 
situations, we can reasonably settle for a 
level which gives us 95 per cent assur-
ance. 

Table 1 shows how the statistical factor 
is computed. Algebraically the "standard 
deviation" is computed by the following 
formula: 

Standard Deviation = S 
~-----=-

v Sum of (A1 - A) 2 
S= N 

By adding three of these standard devi-
ations to the average number of copies 
on loan, we arrive at the number which 
represents the maximum copies that 
would be on loan 95 per cent of the time. 

In summary, the procedure involves 
the following steps: 

1. Compute the average number of cop-
ies on loan. 

2. Compute the average length of time 
on loan. 

3. Compute the standard deviation. 
4. Add the average number of copies on 

loan plus three times the standard de-
viation. 

Now, what does this numerical manip-
ulation do for us? First, as we pointed out 
above, it provides a number by which 
we may judge the adequacy of the num-
ber of copies we own. If we look back 
at our graphic example above, we would 
calculate that 9 5 per cent of the time we 
would have no more than ten copies on 
loan. (See Table 1.) If we do, in fact, 
have more than this number of copies, 
the advisability of procuring more is 
questionable. If we have less copies, the 
acquisition of additional copies is ration-
ally justified, since we expect that we will 

have occasions when we will not be able 
to provide a customer with his requested 
text. If we have a short supply of a large 
number of books, we can even rank them 
and choose those books whose shortage 
is the most critical. 

We must, of course, mention the lim-
itations of this analysis. Like the tool 
crib problem, one critical element in the 
evaluation of this analysis is the effort of 
producing it. The inputs required include 
the detail of the borrowings and returns 
for the desired period of time. Compu-
terization makes the clerical effort feasi-
ble and must be considered an integral 
part of the analysis. The recent develop-
ments reported in the two articles foot-
noted above promise the realization of 
this requirement. 

A second limitation is mathematical in 
nature. If sufficient data is not available 
in relation to the period of observation, 
the results will be distorted. For exam-
ple, if the only activity of a certain book 
during a year were three loans on the 
same day for only one each, we would 
calculate a need for only . 009 books. 
This is an obvious contradiction to the 
facts given in the analysis. We must, 
therefore, impose a requirement that the 
length of observation include that period 
of time in which loans and borrows actu-
ally take place. With this restriction we 
would calculate for three books in this 
last example. Other allowances for which 
adjustments must be made include "lost" 
copies (whether reported in fact or as-
sumed by nonretum), damaged or un-
usable copies, and the like. 

What we have done here is outline a 
path toward a future in library science. 
The contributions to better management 
of libraries will not come easily or cheap-
ly. They must be borrowed and bought 
from a diversity of other fields. The anal-
ysis we have developed here only indi-
cates how a snatch of the rapid technolog-
ical change we see around us can be har-
nessed to a seemingly unrelated field, and 
made to contribute to its advancement. • • 

490 C 0 L L E G E A N D R E S E AR C H L I B R A R IE S