The perception of sensory attributes is often quantified through measurements of discriminability (an observers' ability to detect small changes in stimulus), as well as direct judgements of appearance or intensity. Despite their ubiquity, the relationship between these two measurements is controversial and unresolved. Here, we propose a framework in which they both arise from the properties of a common internal representation. Specifically, we assume that direct measurements of stimulus intensity (e.g., through rating scales) reflect the mean value of an internal representation, whereas measurements of discriminability reflect the ratio of the derivative of mean value to the internal noise amplitude, as captured by the measure of Fisher Information. Combination of the two measurements allows unique identification of internal representation properties. As a central example, we show that Weber's Law of perceptual discriminability can co-exist with Stevens' observations of power-law scaling of perceptual intensity ratings (for all exponents), if one assumes an internal representation with noise amplitude proportional to the mean. We extend this result by incorporating a more general physiology-inspired model for noise and a discrimination form that extends beyond Weber's range, and show that the combination allows accurate prediction of intensity ratings across a variety of sensory modalities and attributes. Our framework unifies two major perceptual measurements, and provides a potential neural interpretation for the underlying representations.