Pacing a Field: An Error Bound for Area Measured by Counting Paces

A rectangular field can be measured with no equipment at all by counting paces along two sides and multiplying, but the accuracy of the result is often overstated. This study separates the error into a random part, which shrinks as the square root of the number of paces, and a systematic part caused by a wrong assumed pace length, which does not shrink at all. Under stated assumptions (mean pace 0.75 m, step-to-step coefficient of variation 0.03 for an ordinary walker, and a 0.01 fractional bias in assumed pace length), a 50 m by 50 m field carries a bias of 2.01% in area and a total error bound of 3.0284% at approximately 95% confidence, while a poorly calibrated walker (coefficient of variation 0.05, bias 0.02) reaches 5.7374% on the same field. The commonly repeated belief that pacing gives area accurate to about one percent does not survive under ordinary assumptions: a pace-length bias of only 0.5% already produces a 1.0025% error in area, and holding a total 1% bound is impossible whenever the bias is 0.01, regardless of field size. A sizing rule is derived: for a ±5% total bound the walker must first reduce bias, because with bias 0.02 and coefficient of variation 0.05 the rule demands 209 paces per side (a side of 156.75 m), whereas a calibrated walker needs only a handful of paces per side for the random part. Calibration on a measured line, repeated, is therefore the controlling step, not walking more carefully over a longer field. This study is simulated, not measured.