This thesis is a feasibility study of using station data and using optimal averaging in spectral space. This method provides a new approach to estimate the global average annual mean temperature directly from station data, instead of going through an intermediate step of data gridding. The method has the potential to reduce the sampling error of the global average. We first show the forward and backward conversion of a function in a spatial space on a 3-dimensional sphere to another function in a spectral space of spherical harmonics. The sampling errors associated with the forward and backward transforms are estimated. This technique is then used to convert the NCEP 20th Century Reanalysis dataset into spectral space and to compute the EOFs. Both normalized and non-normalized representations of the EOFs are displayed to show what information can be gathered from each format. These EOFs are then used to compute the optimal average (OA) of the annual mean surface air temperature data from long-term GHCN (Global Historical Climatology Network) stations. The weight for each station is computed and shows the station's contribution to the global average. Our global average annual mean temperature results are compared with existing results in the climate community. The mathematical theory, the numerical results and the comparison all show that the spherical harmonic method for optimal average of GHCN station data is feasible.