Past methods for studying graph interpretation have only indirectly assessed people’s mental picture of the data that produced the graph. Recently, we developed a more direct, drawing-based measure and used to reveal a severe misinterpretation of the common bar graph of averages: one in five viewers mistook the average for the data's outer limit. Here, we use the same measure to reveal a second misinterpretation, whereby even more viewers—one in three—incorrectly assume that data frequency remains approximately uniform over its entire range. Missing from their mental picture are the tails of the distribution—the relative rarity of extreme values—which are so characteristic of real data that they are embedded in the core normality assumption of statistics. We label this misinterpretation the "Uniformity Fallacy" and characterize its nature, reproducibility, generalizability, and correlates. We conclude that bar graphs of averages fail to communicate data truthfully in not one, but two fundamental ways.