Gene expression depends on the frequency of transcription events (burst frequency) and on the number of mRNA molecules made per event (burst size). Both processes are encoded in promoter sequence, yet their dependence on mutations is poorly understood. Theory suggests that burst size and frequency can be distinguished by monitoring the stochastic variation (noise) in gene expression: Increasing burst size will increase mean expression without changing noise, while increasing burst frequency will increase mean expression and decrease noise. To reveal principles by which promoter sequence regulates burst size and frequency, we randomly mutated 22 yeast promoters chosen to span a range of expression and noise levels, generating libraries of hundreds of sequence variants. In each library, mean expression (m) and noise (coefficient of variation, η) varied together, defining a scaling curve: η(2) = b/m + η(ext)(2). This relation is expected if sequence mutations modulate burst frequency primarily. The estimated burst size (b) differed between promoters, being higher in promoter containing a TATA box and lacking a nucleosome-free region. The rare variants that significantly decreased b were explained by mutations in TATA, or by an insertion of an out-of-frame translation start site. The decrease in burst size due to mutations in TATA was promoter-dependent, but independent of other mutations. These TATA box mutations also modulated the responsiveness of gene expression to changing conditions. Our results suggest that burst size is a promoter-specific property that is relatively robust to sequence mutations but is strongly dependent on the interaction between the TATA box and promoter nucleosomes.