A Cognitive Study Of High School Students' Understanding Of Average

This paper investigated high school students' understanding of the average. The study was divided into 2 parts. It was to carry out the investigation on the understanding of the Average to the high school students having different knowledge backgrounds in the aspect of the knowledge related to statistics. All the subjects were from grades 10, 11, and 12.The problems to be investigated are: (1) when the students summarizing a group of data, whether they know they use Average correctly? If the Average is not suitable, can the students use others such as Medians or Modes properly to represent the average level of the data set? (2) how do the students understand the Weighed Average? (3) how do the students understand the meaning of Average? (4) can the students choose Mathematical Expectation rather than Average in problem solving when there is no hint? Can the 12^th Graders, who have learned Mathematical Expectation, explain the relation between Average and Mathematical Expectation?This investigation includes both questionnaire and selected interviews. Besides testing them, some of them are interviewed. Finally, the following conclusions were obtained:1. Almost all the students have no difficulty in calculating the Average and Weighed Average. Only very few of them made mistakes because of errors in reading.2. when the students are summarizing the Average level of a data set, usually they first use the Average, and with time going on, the frequency of use is gradually increasing, but they are not very clear in which situation the Average is not properly applied and should calculate the Average after removing the extremes or should not use the Average but use other statistics such as Medians, Modes to act as the representatives. But also, they have some difficulties in applying the Weighed Average in problem solving.3 . The high school students understood the Average's "Average" and "Multi-Shift less" meaning and "when one calculates the Average, a value of zero, if it appears, must be taken into account" well but do not understand well of "Average can be no practical meaning" and "the sum of the deviations from the average is zero". Some of the students consider the Average as the middle level, and believe that more than the Average means over the majority, ignoring the nature of "extreme values influence of the Average".4. The students from grade 12 did not understand Mathematical Expectation very well. They could imitate the saying of "Mathematical Expectation reflects the average level of the values of discrete random variables". But they don't really know the relation between the two and just simply use Arithmetic Averages to solve the problems which are supposed to be solved with Mathematical Expectation.In the end, some practical suggestions were proposed to the teaching of Mathematical Expectation and the Average based on the findings of this study. This paper appeals not to be content with the calculation level but should explore to the meaning, nature and application of Average to help students better understand the Average which in primary school we started learning and guide students in the future to better use the tools in hand when they deal with data.