The relationship between the size and the variance of firm growth rates is known to follow an approximate power-law behavior sigma(S) approximately S(-beta(S)) where S is the firm size and beta(S) approximately 0.2 is an exponent that weakly depends on S. Here, we show how a model of proportional growth, which treats firms as classes composed of various numbers of units of variable size, can explain this size-variance dependence. In general, the model predicts that beta(S) must exhibit a crossover from beta(0) = 0 to beta(infinity) = 1/2. For a realistic set of parameters, beta(S) is approximately constant and can vary from 0.14 to 0.2 depending on the average number of units in the firm. We test the model with a unique industry-specific database in which firm sales are given in terms of the sum of the sales of all their products. We find that the model is consistent with the empirically observed size-variance relationship.