# Expected Number of Patients Compromised by Failure

With respect to the window of vulnerability that is opened whenever patient results are reported between the evaluation of QC materials, Parvin and colleagues state that:

- if we consider that a test system failure can begin at any specimen with equal probability, then the expectation is that half the number of patient specimens tested between QC evaluations will be affected in the event of an undetected test system failure.
^{1}

Parvin CA, Yundt-Pacheco J, Williams M. Designing a quality control strategy: In the modern laboratory three questions must be answered. ADVANCE for Administrators of the Laboratory 2011;(5):53-54.

Here is the proof. We begin by assuming that failure is equally likely to occur before any patient specimen between the QC events. If there are n patient specimens between QC events, there are n+1 spaces (we count the space before the last QC event as part of this).

The expected number of patients compromised by failure – or the average number of patients compromised by failure - can be computed as the sum of the patients compromised divided by the number of possible failure locations (there are n+1 failure locations).

If the failure occurs prior to the first patient specimen (but after the QC), then all n patients would be compromised.

If the failure occurs prior to the second patient specimen (but after the first patient specimen), then n-1 patients would be compromised.

If the failure occurs prior to the last patient specimen (but after the second to the last patient specimen), then 1 patient would be compromised.

If the failure occurs after the last patient specimen but before the next QC, then none of the patients would be compromised.

So the sum of the patients compromised is n + n-1 + … + 1 + 0.

This is equal to the sum of the first n integers, which is equal to

.

To compute the expected number of patients compromised by a failure, divide the sum of the first n integers by n+1.

, or ½ the number of patients between QC events.

The expected number of patients compromised by a failure is ½ the number of patients between QC events.

1. Parvin, C.A., Yundt-Pacheco, J., Williams, M. “*Designing a quality control strategy: In the modern laboratory three questions must be answered,” ADVANCE for Administrators of the Laboratory* 2011;(5):53-54.