Let's say that we have a team of twenty, and each of them tracks their time on tasks.
They have a manager who doesn't track time on tasks because his job is to support the team, regardless of what task, plan, mentor and attend meetings. It's not reasonable to allocate his time to tasks.
We calculate the utilization rate for the team by dividing the number of hours they tracks on tasks by the number of hours they logged to payroll. Right now a rate above 50% is good.
If we include the manager in the team utilization, it brings the utilization rate down. This is correct because too many managers to non-managers should yield a poorer utilization. The manager is "dead weight" to the utilization calculation.
In theory a manager can come on board (bringing 8 hours of dead weight per day) but take care of minutia on behalf of the team, this increasing their aggregate tasked hours by more than 8.... a net positive.
This method fails when the manager is sick, or leaves early... or logs fewer payroll hours for whatever reason. If he is sick, then the team sheds 8 hours of "dead weight" but doesn't feel a long-term impact on their own productivity because he was only sick for the day.
How do you account for his time then, when calculating utilization?
I am considering giving him a "free weight" which is equal to the average task hours across the team, then weighted by the ratio of managers to non-managers on team.
Therefore if the team logs an average of 4 hours, then the manager will get 4 * (20/21) = 3.81 hours for free. The result is that the manager always brings down the overall utilization of the team by some margin (so his impact can be visualized on a time-series), and his absence brings it up by a similar margin (but spikes are less severe).