Rational inattention of decision makers to costly data and information and resources affects their optimal decision making strategies. The theory of rational inattention has found applications in several areas such as economics, finance and psychology. In this paper, we study scenarios where the available data is noisy. The noise may have been generated because of inaccuracies or errors in data collection methods, or the data may have been intentionally distorted to protect private or secret information. Here we introduce a formulation for rationally inattentive decision making when the data is noisy, and derive its optimal decision making strategy. Using a stock trading problem as an example, we demonstrate that as the noise level in the data increases, probability of correct decision decreases. This results in less payoff for the decision maker, when using noisy data. We also show how the noise level and information cost parameters can be estimated using the developed formulation. The results are useful for developing decision making strategies, when using noisy data.