= Measurement error
{wiki}
A <measurement error> is the difference between a recorded measurement $W$ and the quantity $U$ intended to be measured. The additive model $W=U+e$ separates that error $e$ from genuine variation in $U$. A shared offset is a <systematic error>; fluctuations between repeated measurements are <random measurement error>. Repeating a measurement can reduce <independent random variables>[independent] <Random measurement error>, but does not remove a shared <systematic error>.
<measurement error> in a <independent variable>[predictor] can bias a <regression coefficient>[regression slope] even when it has <mean> zero. Suppose $Y=a+bU+\varepsilon$, where $e$ is uncorrelated with $U$ and $\varepsilon$, and $\varepsilon$ is uncorrelated with $U$. A regression of $Y$ on the recorded $W$ has population slope
$$
\frac{\operatorname{Cov}(W,Y)}{\operatorname{Var}(W)}=b\frac{\operatorname{Var}(U)}{\operatorname{Var}(U)+\operatorname{Var}(e)}.
$$
Thus equal <independent variable>[predictor] and error <variances> halve the slope: this is <attenuation bias from classical measurement error>. In contrast, <independent random variables>[independent] mean-zero error added only to $Y$ increases residual <variance> without changing that population slope. An <errors-in-variables model> describes the measurement mechanism explicitly rather than treating noisy predictors as exact.
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