Statistical Approach to Defining and Elaborating the numbers behind span of control and quantifying the relationships a People leader manages. It also shows how to calculate the same and has embedded Excel to auto-calculate the Relationships given the number of subordinates one manages.

1. Span of Control –
Quantitative Approach to Relationships
Presented to: Relevant to any People Leader
Date: 09/Nov/2011
Location: NA

2. Agenda – Span of Control, Quantitative Approach
Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships
Definition of:
 Span of control
 Supervisor
 Subordinate
The scope of activities
and inter-relation between a
Supervisor and the
Subordinates
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3. Agenda – Span of Control, Quantitative Approach
Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships
Definition of: Definition of three kinds of
Relationships managed by a
 Span of control Supervisor:
 Supervisor  (DSR) Direct Single
Relationships
 Subordinate
 (CIR) Cross Individual
The scope of activities Relationships
and inter-relation between a
Supervisor and the  (DGR) Direct Group
Subordinates Relationships
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4. Agenda – Span of Control, Quantitative Approach
Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships
Definition of: Definition of three kinds of The Formulas to calculate the
Relationships managed by a three kinds of Relationships
 Span of control Supervisor: that a Manager handles.
 Supervisor  (DSR) Direct Single The basic inputs being the
Relationships number of Subordinates a
 Subordinate Supervisor controls.
 (CIR) Cross Individual
The scope of activities Relationships It has been devised by Graicunas,
and inter-relation between a a Management Consultant in
Supervisor and the  (DGR) Direct Group 1933. We would discuss him at
end.
Subordinates Relationships
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5. Agenda – Span of Control, Quantitative Approach
Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships
Definition of: Definition of three kinds of The Formulas to calculate the There are various ways that
Relationships managed by a three kinds of Relationships the concept of SOC can be
 Span of control Supervisor: that a Manager handles. put to use for enhancing
Management role and
 Supervisor  (DSR) Direct Single The basic inputs being the comparing current and future
Relationships number of Subordinates a workload of Supervisors.
 Subordinate Supervisor controls.
 (CIR) Cross Individual We would try discussing the
The scope of activities Relationships It has been devised by Graicunas, same as case studies.
and inter-relation between a a Management Consultant in
Supervisor and the  (DGR) Direct Group 1933. We would discuss him at
end.
Subordinates Relationships
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6. Span of Control – Definition
Span of Control is defined as the Ratio of Subordinates to one Superior.
In the following sections we would discuss the relationship between the following elements:
 Span of Control
 Relationship Management for a Supervisor
 Potential Relationships a Supervisor has to manage due to the Span s/he is handling
 Quantitative approach to the Relationship Management by a Supervisor
The term ‘Supervisor’ would mean an individual who:
 Leads a group of individuals for a certain purpose (that may change over a period)
 Gets reported ‘directly’ by this group of individuals
 Has definite goals for each individual reporting to her/him and for the group as an entity
The term ‘Subordinate’ would mean an individual who:
 Reports to a particular superior for a particular purpose
 Follows defined timelines set by the superior
 Works towards achieving set goals individually and contributes to the team goals
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7. Relationships a Supervisor handles
Direct Single Relationships (DSR)
The relationship between a Supervisor and individual Subordinates
Sup
Definition (DSR)
A Supervisor has to manage her/his direct relation with each individual
in her/his team. Thus, the Supervisor “Sup” in our example is managing
three relations with his subordinates, “Sub1”, “Sub2” & “Sub3”.
The three relations managed by “Sup” here are:
 DSR1
DSR2
 DSR2
 DSR3
Sub1 Sub2 Sub3
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8. Relationships a Supervisor handles contd…
Cross Individual Relationships (CIR)
The relationship between each individual subordinates, in both ways.
Sup
Definition (CIR)
A Supervisor has to manage the inter-relations between each individual
in her/his team. Thus, the Supervisor “Sup” in our example is managing
six inter-relations within his subordinates, “Sub1”, “Sub2” & “Sub3”.
The six inter-relations managed by “Sup” here are:
 CIR1
 CIR2
 CIR3
Sub2  CIR4
 CIR5
 CIR6
CIR2
CIR5
Sub1 Sub3
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9. Relationships a Supervisor handles contd…
Direct Group Relationships (DGR)
The relationship between Superior and combination of subordinates.
Definition (DGR)
A Supervisor has to manage her/his group-relations between each group
Sub2 Sub1 of individuals in her/his team. The number of groups would be a result
Sub1 Sub2 of each possible combination that can occur with the subordinates.
Sub2 Sub3
DGR9
Thus, the Supervisor “Sup” in our example is managing nine group-
Sup relations within the group of subordinates, Sub1, Sub2 & Sub3.
The nine group-relations managed by “Sup” here are:
Sub1 Sub3
Sub3 Sub1
Sub3 Sub2
Sub3 Sub1 Sub2
Sub1 Sub2 Sub3 Sub2 Sub3 Sub1
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10. Next step after the Theory…
DSR CIR DGR
Calculation & Application
After defining the theory of Relationships a Manager handles, we would discuss the practical aspect of the same. There are two aspects to
Practicality of this Theory:
 Calculating the number of relationships of each kind a Manager handles
 Applying the Theory and the Calculation to Practical life in office
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11. Calculating DSR
Calculating Direct Single Relationships
DSR is the easiest to calculate. It is equal to the number of Subordinates a Supervisor has.
Assumption: Number of Subordinates = n
Formula: DSR = n
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12. Calculating CIR
Calculating Cross Individual Relationships
CIR calculation is not tricky, and the formula finds out the exact number of cross relation a manager handles in
her/his team.
Assumption: Number of Subordinates = n
Formula: CIR = n x (n-1)
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13. Calculating DGR
Calculating Direct Group Relationships
DGR calculation is little tricky, and the formula finds out the exact number of group relation a
manager handles in her/his team.
Assumption: Number of Subordinates = n
Formula: DGR = n x (2n/2-1)
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14. Sum of all Relationships
Total number of relationships
The total number of relationships handled by a Superior would then be a sum of DSR, CIR & DGR. However, there is another formula to calculate the total
number of relationships without calculating DSR, CIR & DGR separately, if required:
The formula is: n*(2n/2+ n -1)
Calculating the figures using an example…
A chart in the embedded Excel File based on the table demonstrates that, as the number of subordinates increases past four, the complexity of the
relationships increases exponentially. This owes primarily to an increase in the number of DGR created by adding a member to an existing group. For
example, adding a fifth subordinate roughly doubles complexity, increasing the total direct plus cross relationships from 44 to 100. Adding a sixth
subordinate more than doubles complexity again, increasing the number of relationships from about 100 to 222. For 12 subordinates, the total number of
relationships that might demand a superior's attention is an astounding 24,708!
To view the live example of above theory, please open the embedded Excel file at right to have the formulas
and calculations already set up for you in an Excel File…
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15. Practicality & Applicability
“There is nothing which rots morale more quickly and more completely than poor communication and indecisiveness -- the feeling that those in authority do not know their own
minds. And there is no condition which more quickly produces a sense of indecision among subordinates or more effectively hampers communication than being responsible to a superior
who has too wide a span of control”
- Lyndall F. Urwick (Pg 43 of article, "The Manager's Span of Control" published in May-June 1956 issue of Harvard Business Review)
Practicality of quantifying SOC
 SOC affects the cost
A wider SOC, as in the above statement, at times creates unrest among the subordinates. However, a wider SOC ensures lower cost of management. Though, the
significance lies is comparing the cost of poor quality of output of the team against cost of maintaining thinner SOC. Because, as a thumb rule, thinner SOC ensures lesser number of
relations that a superior manages and therefore, efficient output.
 SOC affects subordinate morale
As discussed above, a wider SOC at times creates unrest in subordinates as the superior has a visible span of large number of subordinates and it exposes her/his ability
to do People Management. Even a Super Efficient supervisor would have a hard time to handle 10 subordinates where s/he is actually handling a total of 5210 relationships
(DSR+CIR+DGR).
 SOC behaves typically for different industries
Affording a wider or a thinner SOC depends on the kind of industry as well. In an industry where the output is co-dependent on machines (for e.g. manufacturing), a
wider span of control can be afforded. However, in service industry as ours, thinner SOC results better as in service industries, the output is dependent on the human resource only.
Therefore, managing human relations and inter-relations are more significant
 SOC varies with hierarchy
For any kind of industry, the general approach should be thinning the SOC towards the bottom of hierarchy. The justification being, at the top of hierarchy, the superior
has to manage fewer number of ancillary tasks and major number of human relations. Thus, the people management becomes easier. However, at the lower levels, the transaction
management also becomes as important as human resource management. Thus, managing the total number of relationships (DSR+CIR+DGR) becomes difficult.
SOC is highly dependent on the culture of the kind of organization and industry. It can be used judiciously to control the cost and output of the organization and define the optimum mix so
that there is a maximization in earning.
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16. Graicunas.
The Graicunas formula is named after V.A. Graicunas, who gave a mathematical formula to explain the complexity of span of control if
more subordinates are added to the executive. Every executive 'always measures the burden of his responsibility to control the subordinates in
terms of single relationship between himself and his subordinates.
Graicunas feels that in any group, the relations between executive and his subordinates cannot just be calculated based on single relationship
alone. According to him, there also exists cross relationships which increase in mathematical proportion. The direct single relationship always
increase in the same proportion as the number of subordinates. In such a case each addition to t h ~ I , group would only create a single direct
relationship. But according to Graicunas there also exists direct group and cross relationships which increase very rapidly than the 1I1 increase in
the number of subordinates. This is n~a because the addition of each !/ individual results in many cross and drrett group relationships as there
are persons already in the group. Therefore, the number of relationships increases in exponential proportion. Based upon his studies, Graicunas
develops a formula to calculate the /I/ number of relationships to enable the executives to-examine the complexity of span of control.
According to these formula, in any organization if there are three subordinates direct single relationships would be three, cross relationships six
and direct group relationships nine. But if one more member is added there would nor be any change in the direct single relationships which would
be four, but the cross relationships would increase to 12 and direct group relationship, however, rise exponentially to 28. This explains that
addition of each member to the group under the control of the executive would increase the number of direct group relationships to such an extent
that direct contribution becomes difficult in some cases even impossible. Graicunas also added that for subordinates it is clhite easy to grasp and
remember every combination of groups. But from five on, it is not possible to remember because the relationships become more of confusion.
When the x principle enunciated by Graicunas is valid or not, whether the formulae has empirical validity or not, the problem that any increase in
the number of subordinates would lead to complexity in the relationships between the individual and groups has aptly been brought out by
Graicunas.
It is this factor that needs to be carefully considered in any discussion on how many subordinates an executive can effectively control.
www.nickols.us/graicunas.pdf
http://en.wikipedia.org/wiki/Graicunas_formula
http://en.wikipedia.org/wiki/Span_of_control
www.jstor.org/stable/254987
www.bus.lsu.edu/bedeian/articles/VGracunias-AMJ1974.pdf
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17. K. S. Alok Ranjan
+91 9818665496 | +91 124 4118792
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