Combining multiple information sources such as subbands, streams (with different features) and multi modal data has shown to be a very promising trend, both in experiments and to some extend in real-life biometric authentication applications. Despite considerable efforts in fusions, there is a lack of understanding on the roles and effects of correlation and variance (of both the client and impostor scores of base-classifiers/experts). Often, scores are assumed to be independent. In this paper, we explicitly consider this factor using a theoretical model, called Variance Reduction-Equal Error Rate (VR-EER) analysis. Assuming that client and impostor scores are approximately Gaussian distributed, we showed that Equal Error Rate (EER) can be modeled as a function of F-ratio, which itself is a function of 1) correlation, 2) variance of base-experts and 3) difference of client and impostor means. To achieve lower EER, smaller correlation and average variance of base-experts, and larger mean difference are desirable. Furthermore, analysing any of these factors independently, e.g. focusing on correlation alone, could be miss-leading. Experimental results on the BANCA and XM2VTS multi-modal databases and NIST 2001 speaker verification database confirm our findings using VR-EER analysis. Furthermore, F-ratio is shown to be a valid criterion in place of EER as an evaluation criterion. We analysed four commonly encountered scenarios in biometric authentication which include fusing correlated/uncorrelated base-experts of similar/different performances. The analysis explains and shows that fusing systems of different performances is not always beneficial. One of the most important findings is that positive correlation ``hurts'' fusion while negative correlation (greater ``diversity'', which measures the spread of prediction score with respect to the fused score), improves fusion. However, by linking the concept of ambiguity decomposition to classification problem, it is found that diversity is not sufficient to be an evaluation criterion (to compare several fusion systems), unless measures are taken to normalise the (class-dependent) variance. Moreover, by linking the concept of bias-variance-covariance decomposition to classification using EER, it is found that if the inherent mismatch (between training and test sessions) can be learned from the data, such mismatch can be incorporated into the fusion system as a part of training parameters.