The statistical significance that characterizes a discrepancy between a measurement and theoretical prediction is usually calculated assuming that the statistical and systematic uncertainties are known. Many types of systematic uncertainties are, however, estimated on the basis of approximate procedures and thus the values of the assigned errors are themselves uncertain. Here the impact of the uncertainty {it on the assigned uncertainty} is investigated in the context of the muon $g-2$ anomaly. The significance of the observed discrepancy between the Standard Model prediction of the muons anomalous magnetic moment and measured values are shown to decrease substantially if the relative uncertainty in the uncertainty assigned to the Standard Model prediction exceeds around 30%. The reduction in sensitivity increases for higher significance, so that establishing a $5sigma$ effect will require not only small uncertainties but the uncertainties themselves must be estimated accurately to correspond to one standard deviation.
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