On the Effect of Decimal Corrections on Errors of Observation

Abstract

Let $t$ be the true value of what is being measured and suppose that the error of observation is a symmetric normal distribution of standard deviation $\sigma$. The "rounding-off" error due to the reading of measurements to the nearest unit has a distribution and an expected value depending on $t$ and $\sigma$. It is shown that, for a fixed $\sigma > 0$, the expected value of the decimal correction, $r(t; \sigma)$, is an analytic function of $t$ which is odd, of period 1, positive for $0 < t < \frac{1}{2}$, and has a convex arch as its graph on $0 \leqq t \leqq \frac{1}{2}$. Furthermore, if $0 < t < \frac{1}{2}$, both $r(t; \sigma)$ and its maximum value, $\operatorname{Max}_t r(t; \sigma)$, are decreasing functions of $\sigma$.