A note on calculating the average span of control

Paper highlights

Average span of control summarizes how many direct reports each supervisor has in an idealized hierarchy. A widely cited formula from Blau and Scott’s Formal Organizations calculates it incorrectly when the available inputs are an organization’s total employees and number of supervisory levels. This note derives the correct calculation.

The distinction matters most in small organizations or tall hierarchies, where supervisors make up a substantial share of the workforce. Treating the total workforce as if everyone were in the lowest layer then produces a sizable overestimate.

The calculation

If every supervisor has the same span S and the organization has L supervisory levels, total employment N is the sum of the employees at every level: one person at the top, S at the next level, then S squared, continuing through level L. Average span is the value of S that makes this geometric sum equal N.

There is no general closed-form expression for S from N and L, so it must be solved numerically. The older formula instead works when its employment input is the number of people in the bottom layer, not the total number of employees.

When it matters

• The older formula overstates span by treating all employees as if they occupied the bottom layer; the error is largest in small or tall organizations.
• The correction supports comparisons using the aggregate quantities datasets commonly report: total employees and supervisory levels.
• When person-level reporting data exist, researchers should calculate actual spans rather than impose equal branching.

The corrected measure remains a summary of an idealized hierarchy. Uneven spans, dual reporting lines, vacancies, and non-supervisory specialists require more detailed measures.

How to use this note

Cite this for

• The corrected calculation of average span of control from total employees and supervisory levels.
• The distinction between total employment and the number of people in the bottom layer of an idealized hierarchy.
• A concise measurement correction for organization-design research using aggregate hierarchy data.

Useful for teaching

• Why a plausible organizational formula can be wrong for the data researchers actually observe.
• How idealized hierarchy measures differ from actual person-level reporting structures.
• Why small and tall organizations are most sensitive to this measurement error.

Careful claim

The note corrects span-of-control calculation for an idealized equal-branching hierarchy when only total employment and supervisory levels are known; use actual reporting-line data when available.

Other links

• SSRN preprint

Last updated 2026-06-21