  The first part of our analysis uses the wavelet method to compare the Quantum
Chromodynamic (QCD) prediction for the ratio of hadronic to muon cross sections
in electron-positron collisions, $R$, with experimental data for $R$ over a
center of mass energy range up to about 7 GeV. A direct comparison of the raw
experimental data and the QCD prediction is difficult because the data have a
wide range of structures and large statistical errors and the QCD description
contains sharp quark-antiquark thresholds. However, a meaningful comparison can
be made if a type of "smearing" procedure is used to smooth out rapid
variations in both the theoretical and experimental values of $R$. A wavelet
analysis (WA) can be used to achieve this smearing effect. The second part of
the analysis we concentrate on the 3.0 - 6.0 GeV energy region containing the
relatively wide charmonium resonances $\psi(1^-)$. We use the wavelet
methodology. Both analyses are examples of the usefulness of WA in extracting
information in a model independent way from high energy physics data.
