During the past few years, several instruments for color measurement have become commercially available at unprecedented low prices. Although these instruments are no spectrophotometers, their price-performance ratio may be attractive for applications that do not require the high accuracy of traditional instruments. We investigated the performance of this class of instruments. These instruments typically do not provide spectral reflectance data, and some do not even produce colorimetric data such as CIELab coordinates. Therefore, in this investigation we express performance of the devices as the percentage of correctly identified color chips in color documentations (fan decks) such as those from RAL and Dulux CP5. Based on statistical physics, we develop a mathematical model that predicts the performance of a low-cost color instrument. This performance is shown to depend mainly on (i) the measurement precision of the instrument, and on (ii) the density of chip color coordinates in the color documentation. The new model makes it possible to compare test results from different instruments, even if they were obtained with different color documentations. The model also can be used to help design an efficient quality control test for the instruments, and to design new color documentations for which the instrument gives superior performance.