We have applied so-called `optimised' perturbation theory to resolve
the renormalisation-scale (<i>mu</i>) ambiguity of exact 
 O(<i>alpha</i><sup>2</sup><sub>s</sub>) 
 QCD calculations of event shape observables in e<sup>+</sup>
e<sup>-</sup> hadrons. We fitted the optimised predictions for 15
observables to hadronic <i>Z</i><sup>0</sup> decay data from the SLD
experiment to determine
 <i>alpha</i><sub>s</sub>(<i>M<sup>2</sup><sub>Z</sub></i>). 
 Comparing with results using the physical scale <i>mu</i> =
<i>M<sub>Z</sub></i> we found no reduction in the scatter among
 <i>alpha</i><sub>s</sub>(<i>M<sup>2</sup><sub>Z</sub></i>)
values from the 15 observables, implying that the 
 O(<i>alpha</i><sup>2</sup><sub>s</sub>) 
 predictions with optimised scales are numerically no closer to the
exact all-orders results than those with the physical scale.