Borrowing the Bathroom Scale: Difference Weighing of an Infant and the Averaging Rule That Makes a Week's Growth Visible

An infant can be weighed on an ordinary bathroom scale by difference: the caregiver steps on alone, then steps on holding the infant, and subtracts. This paper computes the error budget of that method for two household scale classes, from stated assumptions about resolution, repeatability, stance variation, and infant movement, and derives the averaging rule that makes a week's growth visible. A single difference weighing has a standard deviation of 162.7 g on a digital scale with 100 g resolution, and 372.9 g on a dial scale with 500 g divisions, so a single pair of weekly sessions carries a weekly-change standard deviation of 230.1 g and 527.4 g respectively — larger, in both cases, than the 150 g weekly gain the weighing is meant to detect. Averaging repeated difference weighings under stated conditions (same time of day, before a feed, dry nappy) repairs the digital scale: 11 repeats per session detect a true gain of 150 g per week at the stated 95% criterion, inside a feasibility cap of 12 repeats, and at the cap the smallest detectable gain is 141 g. It does not repair the dial scale, which would need 55 repeats per session to detect the same gain — beyond any feasible cap — and whose smallest detectable gain at the cap is 303 g per week. Under uncontrolled conditions, with weighings taken at random times relative to feeds, no number of repeats detects 150 g per week on either scale, because the between-session variation in gut and bladder contents does not average away within a session. Conditions come before repetition, and repetition before interpretation: a weekly change read from one difference weighing on any household scale is noise, and a dial-scale household should track growth by other signs and by clinic weights rather than by difference weighing.