The understanding of children and adults regarding four component properties of the arithmetic mean was studied. The component properties were: (1) the mean is a data point between extreme values of a score distribution; (2) the sum of deviations about the mean equals zero; (3) when the mean is calculated, any value of zero must be taken into account; and (4) the average value represents the values that were averaged. Also studied were the relative difficulties of these properties, the potential of varied problem formats to facilitate understanding of these concepts, and differences between methods of investigating understanding (individualized oral tests and large-group paper-and-pencil tests). A 16-item test that presented the properties in story and numerical form was completed by 41 fourth graders, 60 eighth graders, and 40 college students. Data were analyzed quantitatively and qualitatively. Mastery of all properties increased with age. Story format was superior to numerical format, and the first two properties were significantly easier to master than the latter two, possibly because of an artifact of item format. Differences between individualized oral tests and paper-and-pencil methods can cause varying conclusions to be drawn about subjects' understanding of the arithmetic mean. (SLD)