In this article, an attempt has been made to compare two methods of determining potential span of control, that is, the Stieglitz method and the Graicunas method. It appears that the application of those two approaches leads to similar ramifications. They both have essential failures, which fact leads to the conclusion that new and improved methods for determining the optimal number of subordinates for a given manager must be sought. This results in the more effective decision-making processes.Stieglitz method, Graicunas method, span of control, number of subordinates

Janusz Martan

Marian Hopej

English

Research Papers in Economics

BADANIA OPERACYJNE I DECYZJE
Nr 2 2006
Marian HOPEJ*,
Janusz MARTAN**
THE DETERMINATION OF SPAN OF CONTROL
In this article, an attempt has been made to compare two methods of determining potential span
of control, that is, the Stieglitz method and the Graicunas method. It appears that the application of
those two approaches leads to similar ramifications. They both have essential failures, which fact
leads to the conclusion that new and improved methods for determining the optimal number of sub-
ordinates for a given manager must be sought. This results in the more effective decision-making
processes.
Keywords: Stieglitz method, Graicunas method, span of control, number of subordinates
In the science of organization and management, three types of span of control can
be distinguished, namely, potential, formal and real span of control. Potential span of
control is the number of subordinates a manager can effectively manage in given con-
ditions. Formal span of control refers to the number of people directly and formally
(in formal organizations) subordinate to a manager. Real span of control, on the other
hand, measures the number of subordinates who in fact directly report to the manager
[Kieżun, 1999], [Pszczołowski, 1978], [Zieleniewski, 1974].
Some significant discrepancies may appear between the above-mentioned spans of
control. When potential span of control is bigger than formal and real ones, the man-
ager has the feeling that he does not have enough work to do. When formal and real
spans of control are bigger than potential one, the manager is not able to formally and
really supervise the subordinates. Some of the employees may get out of his influence
and an informal deputy manager may even emerge. Hence, real and formal span of
control should agree with potential span of control [Hopej, 2005].
                                                     
* Institute of Engineering and Management, Wroclaw University of Technology, ul. Smoluchow-
skiego 25, 50-372 Wrocław, e-mail: marian.hopej@pwr.wroc.pl
** Institute of Applied Information, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27,
50-370 Wrocław, e-mail: janusz.martan@pwr.wroc.plM. HOPEJ, J. MARTAN 56
In theory and practice four methods of calculating potential span of control are
known. They are as follows: analogy, intuitional, mathematical and factorial method
[Kieżun, 1997].
According to the oldest Old Roman sources the rule of the ten-person span, con-
stituting the basis of the organization of the legions, was based on the analogy of the
number of fingers. Napoleon, who was widely known for keeping one of his hands in
his breast pocket, was in favour of the five-person span, whilst H. Fayol intuitively
determined the potential span of control to be 15 [Kieżun, 1997].
Graicunas, on the other hand, followed a different path and distinguished three
types of relationships between superior and subordinates:
• Direct single relationships between superior and individual subordinates. The
number of possible direct individual relationships  therefore equals the number of
subordinates n.
• Direct group relationships. Graicunas assumes that in contacts between a supe-
rior and a group of subordinates, one of the subordinates comes always to the fore.
This means that in case of two subordinates (B, C), there are two groups BC and CB
(Fig. 1).
A B C A B C
Fig. 1. Relations between superior and a group of subordinates
The following is taken from the author’s own research
Hence, the total number of relationships amounts to:
) 1 2 (
1 − =
− n n X (1)
where X is the number of interrelationships between the superior and his subordinates
and n is the number of subordinates.
• Relationships between subordinates, which absorb the manager’s attention.
Only pairs with one person as an initiator were taken into consideration (Fig. 2).
BC CB
Fig. 2. Relations between the subordinates
The following is taken from the author’s own researchThe determination of span of control 57
The number of interrelationships between the subordinates can be calculated
from the formula:
Y = n(n – 1) (2)
where Y is the number of relationships between subordinates and n is the number of
subordinates.
If we add all the types of relationships, we can calculate their total number Z from
the formula:
) 1 2 ( ) 1 ( ) 1 2 (
1 1 − + = − + − + =
− − n n n n n n Z
n n (3)
It follows from the formula that each rise in the number of subordinates by one
more than doubles the number of possible interrelationships. With three subordinates
the number of interrelationships is 18, with four it is 44, five bring the total up to 100,
whilst six make a total of 222.
Graicunas’ idea is often referred to, especially by Polish authors, as an example of
basic mistake. “One cannot find two identical, addable social interactions, identical
managers, subordinates or organisational situations. Therefore, the determination of
potential span of control cannot be the result of abstract analysis but it should result
from the study of many interrelated factors” [Koźmiński et al., 1995].
Consequently, it is generally assumed that the determination of potential span of
control must be the result of the study of various factors. Polish author Zieleniewski
distinguished two groups of factors which influence the number of subordinates who
can be directly and adequately managed. The first group comprises:
• the complexity of the decisions made by the manager and connected with it, the
necessary time needed to prepare and make decisions,
• the load on the superior with non-managerial activities (executive ones),
• the load on the superior with external contacts.
The second group consists of:
• the abilities of superiors and the resulting degree of their independence in deci-
sion making and undertaking their actions,
• the amount and nature of interdependence between subordinate posts or organ-
isational units and spatial spread of subordinates,
• the complexity and repeatability of actions taken by subordinates.
As emphasised by the above mentioned author, the more time the superior must
spend on contacts with subordinates, the bigger the potential span of control. These
contacts, on the other hand, become less time consuming as the abilities of the subordi-
nates grow, relationships between the positions become weaker, the spatial spread
shrinks and the activities of the subordinates become more routine [Zieleniewski, 1974].
Another attempt to determine the potential span of control is the Stieglitz method,
favourably evaluated because of the possibility of its practical application. It is basedM. HOPEJ, J. MARTAN 58
on the assumption that span of control depends on seven factors, for which “values”
were established, according to common sense and experience (Table 1).
Table 1
The degree of the load on the superior as a function of the intensity
of the factors of potential span of control
Factor The degree of the load on the superior
Identical Strikingly
similar
Similar Fundamentally
different
Completely
different
The similarity
of the functions
12 2 4 5
all together all in one
building
on the premises spread in the city spread in a coun-
try
Spatial spread
of subordinates
11 2 4 5
Simple,
recurring
Routine Complicated Very complicated Very complicated
and diverse
The complexity
of functions
24 6 8 1 0
minimum of
hints and
control
limited
supervision
periodical su-
pervision and
control
frequent system-
atic control
frequent system-
atic and strict
control
Staff qualifica-
tions
26 9 1 2 1 5
minimun
contacts
limited
contacts
more contacts,
easy coordina-
tion
permanent con-
tacts in case of
serious problems
broad unrepeat-
able contacts
Coordination
24 6 8 1 0
 minimum limited
range
bigger range considerable
range
wide range exclu-
sively for politics
Planning
24 6 8 1 0
Taken from J. Jaszek 1984, p. 176.
Next, on the basis of empirical-contrastive analysis the number of subordinates
was assigned to point ranges which describe the influence of selected factors on the
number of subordinates, constituting the potential span of control (Table 2).
Table 2
The load on the superior as a function of the number of subordinates
Load in points 40–42 37–39 34–36 31–33 28–30 25–27 22–24
The number of
subordinates 4–5 4–6 4–7 5–8 6–9 7–10 8–11
Taken from J. Jaszek, 1984, p. 175.The determination of span of control 59
The theories described above are independent; there is no mention of their inter-
relationship in the related literature. It would be interesting to present the load on the
superior as a function of the number of relationships between superior and subordi-
nates; in other words, it would be interesting to link the Graicunas theory with the
Stieglitz theory.
Figure 3 shows the relationships between the degree of the load on the superior
as a function of the degree of intensification of the factors of potential span of con-
trol, for six of the factors proposed by Stieglitz. As easily noticed, the relationships
under consideration are linear functions and can be approximated by analytical
formulas.
123
 
45
1
3
5
7
9
11
13
15
the degree of the intensification 
of the factors of potential span  of control
     
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   spatial spread of the functions        (2)
   the complexity of the functions        (3)  
 staff  qualifications                           (4)
 
 coordination                                     (5)
 Planning                                          (6)
similarity of the functions                 (1)
t
h
e
 
d
e
g
r
e
e
 
o
f
 
t
h
e
 
l
o
a
d
o
n
 
t
h
e
 
s
u
p
e
r
i
o
r
 
(
i
n
 
p
o
i
n
t
s
)
(3, 5, 6)
(4)
(1, 2)
Fig. 3. The impact of the degree of intensification of the factors of potential span
of control on the load on the superior
The following is taken from the author’s own research
In Figure 4, the impact of the number of subordinates on the load on the superior
(graphical interpretation of Table 2) for two extreme numbers of subordinates is pre-
sented.
The relations presented in Figure 4 were approximated by a straight line
n = 16,72 – 0,312·x (4)
where n is the number of subordinates and x is the load on superior.
planning
(1)
(2)
(3)
(4)
(5)
(6)M. HOPEJ, J. MARTAN 60
15   20   25   30   35   40   45  
2  
4  
6  
8  
10 
12 
 
-
 the upper limit of the range of the number 
of subordinates
 
the lower limit of the range   
of the number of subordinates   y=16,72- 0,13x
load on the superior (in points)
n
u
m
b
e
r
 
o
f
 
s
u
b
o
r
d
i
n
a
t
e
s
Fig. 4. The impact of the number of subordinates on the load on superior
The following is taken from the author’s own research
Putting the above expression into the Graicunas formula, we get the impact of the
number of relationships between superior and subordinates on the degree of the load
on the superior (see Figure 5) expressed analytically by the formula
F = 16,408·2
15,72–0,312·x + 257,94 (5)
where F is the number of relationships between the superior and subordinates and x is
the load on superior.
The relationship analysed is of exponential  character, which means that even
a small change in the number of relationships between superior and subordinates in-
fluences in a fierce way the degree of the load on the superior. As one can see, the
attempt to link two allegedly unrelated theories was successful.
1000 10000 100000
5
10
15
20
25
30
35
40
45
the number of relationships 
between superior and subordinates
l
o
a
d
 
o
n
 
s
u
p
e
r
i
o
r
 
(
i
n
 
p
o
i
n
t
s
)
Fig. 5. The impact of the load on the superior on the number of relationships
between superior and subordinates
The following is taken from the author’s own research
The analysis carried out could have indicated that the criticism of Graicunas’ idea
is of little legitimacy. It does not do this, however, because it is difficult, if not impos-The determination of span of control 61
sible, to assume that all three types of relationships are of the same significance. Inter-
relationships should especially be evaluated in a different way than direct relation-
ships. Moreover, the assumption that each attempt to exert an influence on someone
absorbs the same amount of time is very doubtful.
Additionally, there are also some doubts concerning the Stieglitz method. One
may claim that it does not take into account the greater number of factors, such as
time pressure or organisational culture. Furthermore, the way of assigning values to
individual factors is also debatable.
Hence, if the application of the Stieglitz method and the Graicunas method leads
to similar consequences, none of them should constitute the basis for the determina-
tion of span of control in specific organisations. However, it is necessary to determine
the span of control, because resulting from the rule of the limited span of control,
multidimensional hierarchies are still “the best mechanism we have at our disposal to
do a complex work. It would be unrealistic to expect that we will get rid of them in
a foreseeable future. The wiser solution would be the acceptance of the fact that hier-
archies exist and will always exist, and making an effort to limit their poisonous side
effects as well as causing that the life of people in those hierarchies will be better,
taking care at the same time of workers’ efficiency” [H.J. Leavitt, 2004, p. l18].
What, is therefore, the way of determining the span of control? We suggest (in
broad outline) an approach a little bit different from those most often met in the lit-
erature of the subject. In our opinion, span of control is an endless process. This
means that in creating an organisational structure, one should, by the use of factorial
method encompassing as many factors as possible, determine potential span of control
and put the findings into practice. Next, the superior should observe carefully the
behaviour of subordinates and listen intently to their needs and problems. Such obser-
vation enables the correction of the previously made decision, which may be wrong
because of the difficulties with predicting human behaviour.
Therefore, the correcting mechanism is necessary when determining the span of
control. This also results from the fact that factors influencing the span of control,
because of their dynamic character, undergo frequent changes. As a consequence, it is
essential to monitor constantly the implemented solution, verifying its relevance in
new, altered conditions [M. Czerska, 1998]. This gives a stimulus to the disclosure of
possible need of another change.
References
[1] CZERSKA M., Jak przeprowadzić zmiany organizacyjne w firmie, Ośrodek Doradztwa i Doskonalenia
Kadr, Gdańsk 1997.
[2] HOPEJ M., O rozpiętości kierowania [in:] Eseje, Lesław Martan (red.), Oficyna Wydawnicza Politech-
niki Wrocławskiej, Wrocław 2005.
[3] JASZEK J., Doskonalenie struktury organizacyjnej przedsiębiorstwa, TNOiK, Katowice 1984.M. HOPEJ, J. MARTAN 62
[4] KIEŻUN W., Sprawne kierowanie organizacją, Szkoła Główna Handlowa, Warszawa 1997.
[5] KOŹMIŃSKI A.K., PIOTROWSKI W. (ed.), Zarządzanie. Teoria i praktyka, Wydawnictwo Naukowe
PWN, Warszawa 1995.
[6] LEAVITT H.J., Dlaczego hierarchie kwitną, Harward, Business Review, wrzesień 2003.
[7] PSZCZOŁOWSKI T., Mała encyklopedia prakseologii i teorii organizacji, Ossolineum, Wrocław–War-
szawa–Kraków–Gdańsk, 1978.
[8] STEINMANN H., SCHREYÖG G, Zarządzanie, Oficyna Wydawnicza Politechniki Wrocławskiej, Wro-
cław 1995.
[9] ZIELENIEWSKI J., Organizacja i zarządzanie, PWN, Warszawa 1974.
Ustalanie rozpiętości kierowania
W artykule podjęto próbę porównania dwóch metod ustalania potencjalnej rozpiętości kierowania, tj.
metody Graicunasa i Stieglitza. Okazało się, że ich stosowanie prowadzi do zbliżonych konsekwencji.
Obie metody mają istotne wady i należy poszukiwać innych, doskonalszych metod ustalania właściwej
liczby bezpośrednich podwładnych określonego kierownika, co prowadzi do sprawniejszego procesu
decyzyjnego.
Zaprezentowany zarys nowej metody opiera się na założeniu, iż ustalenie rozpiętości kierowania jest
procesem, który nigdy się nie kończy. Musi być w niego wbudowany mechanizm korygowania, ponieważ
ciągle zmieniają się warunki pracy zarówno przełożonego, jak i podwładnych. Istnieje więc w praktyce
konieczność stałego monitorowania wdrożonego rozwiązania pod kątem dalszej jego aktualności w no-
wych, zmienionych warunkach. Daje to asumpt do ujawnienia się ewentualnej potrzeby kolejnej zmiany
organizacyjnej.
Słowa kluczowe: metoda Stieglitza, metoda Graicunasa, zakres decyzyjny, liczba podwładnych

The determination of span of control

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