Timing the Piece Rate: Observer Error and Rounding Bounds on the Stopwatch Standard Behind Piecework Pay

Piecework pay rests on a standard time: a worker is paid a fixed amount per piece, set by timing the job and converting that time at an hourly rate. This paper computes how errors in the timing propagate into pay. Three timing errors are modelled — stopwatch observer error, sampling variation between cycles, and rounding of the standard — under stated conditions: a single button-press standard deviation of 0.12 s, a systematic bias bound of 0.10 s per timed interval, and true cycle-to-cycle variation of 8% of cycle time. The central result is that the observer's hands are not the main problem. For a 30 second task timed over 10 cycles, observer error contributes only 0.18% while sampling contributes 2.53%, giving a combined random standard deviation of 2.54%, a 95% random bound of ±4.97%, and, with bias and rounding to the nearest 0.01 minute, a worst-case stated bound of 6.43% — about $1.16 per hour on an $18.00 hourly base. Rounding is not trivial either: rounding to the nearest 0.01 minute adds a standard deviation of 0.58% on a 30 second task, and systematically rounding up instead of to nearest donates a further 1.00%. A protocol that bounds the total error to a strict 2.0% at the 95% level requires about 91 cycles for a 30 second task with electronic timing to 0.1 s — far more than the handful of cycles commonly taken. A practical 30-cycle electronic protocol bounds the error to about 3.2%. Standards set from small samples should therefore be published with their bound, and pay disputes about a percent or two are, on these numbers, disputes inside the measurement noise unless many cycles were timed. AI disclosure: Dave, an AI research system, assisted in the preparation of this work.