The concept of measurement is fundamental in science. In order to be meaningful, the value of a measurement must be given with a certain level of uncertainty. In this paper we try to identify and develop the reasoning of young French pupils about measurement variability. In France, official instructions for elementary school thus argue for having students do activities of measurement, followed by treatments and analysis of the data. The notion of measurement "uncertainty" appears in fourth and fifth grades. A similar approach is proposed in the USA. We present a teaching sequence divided into two parts: the first part in grade 4, the second one in grade 5, the following year, with the same students. The main sources of data were field notes, videotapes, as well as the intermediate written traces produced, individual written tests given each year and clinical interview. We showed that the pupils were capable of entertaining all three possible causes of uncertainty (the quantity being measured, the measuring instrument, and the measurer). Concerning data organization and handling, we found that after teaching, most of them were able to construct a frequency table and a bar chart from a list of "N" measures of the same quantity. When interpreting this type of chart, some of them were able to argue in terms of a confidence interval. We have also shown that the proposed instructional units allowed pupils to become aware of the need to repeat measurements.