Making use of EOF analysis and statistical optimal averaging techniques, the problem of random sampling error in estimating the global average temperature by a network of surface stations has been investigated. The EOF representation makes it unnecessary to use simplified empirical models of the correlation structure of temperature anomalies. If an adjustable weight is assigned to each station according to the criterion of minimum mean-square error, a formula for this error can be derived that consists of a sum of contributions from successive EOF modes. The EOFs were calculated from both observed data and a noise-forced EBM for the problem of one-year and five-year averages. The mean square statistical sampling error depends on the spatial distribution of the stations, length of the averaging interval, and the choice of the weight for each station data stream. Examples used here include four symmetric configurations of 4 × 4, 6 × 4, 9 × 7, and 20 × 10 stations and the Angell-Korshover configuration. Comparisons with the 100-yr U.K. dataset show that correlations for the time series of the global temperature anomaly average between the full dataset and this study's sparse configurations are rather high. For example, the 63-station Angell-Korshover network with uniform weighting explains 92.7% of the total variance, whereas the same network with optimal weighting can lead to 97.8% explained total variance of the U.K. dataset.