Infrared and collinear safe event shape distributions and their
mean values are determined at centre-of-mass energies between 45
to 202GeV.
A phenomenological analysis in view of power correction models
including hadron mass effects for both, differential distributions
and mean values is presented.
Using power corrections alpha_s is extracted from mean values
and shapes. As an alternative approach renormalisation group
invariance (RGI) is used as an explicit constraint, leading to a
consistent description of mean values without need for sizeable
power corrections. The QCD beta-function is precisely measured
using this approach. From the DELPHI data on 1-Thrust including
data of low energy experiments one obtains
beta_0 = 7.86 +- 0.32
for the one loop coefficient of the beta-function or, assuming
QCD for the number of active flavours
n_f = 4.75 +- 0.44
These values agree well with QCD expectation, beta_0=7.67 and
n_f=5.