Abstract:

The concept of sampling is receiving increasing attention from mathematics education research, since ideas linked to sampling underlay the work with simulation, which is currently recommended to improve understanding of probability and statistical inference. Moreover, this concept establishes a bridge between statistics and probability and plays a key role in the study of topics such as the frequentist approach to probability or the law of large numbers.

In the Spanish curricular guidelines, the concepts of population and sample appear in the first two grades of secondary education. During the third grade, students learn different methods of collecting samples and the idea of representativeness. However, previous research carried out in other countries suggests that students do not perceive the sampling variability.

The aim of this paper is to analyse Spanish secondary school students’ intuitive understanding of the relationships between the population proportion and the expected value of a sample proportion, as well as its variability in different samples. The sample is composed of 302 secondary school students, 157 from 2nd grade (12-13-year-olds) and 145 of 4th grade (15-16 year-olds) from two different public schools of Huesca who had studied some elements of probability in the previous years.

These students were given a questionnaire with sampling tasks. In this paper, we analyse the results for one item, where students were asked to write four probable values for the number of heads when throwing 100 coins. The mathematical model implicit in this situation is the binomial distribution with parameters n=100 (sample size) and p=0.5 (population proportion for the event in which we are interested). We performed a statistical analysis of the four responses provided by each student and compared them by group. The average value of the four values indicated by the students was used to evaluate their intuitive understanding of the relationship between the population and sample proportions and the range of these four values served to assess their intuitive understanding of sampling variability.

The statistical analysis of this data suggests a good understanding of the relationship between the population and sample proportions, given the close proximity between the theoretical proportion in the population (0.5) and the mean value of the distribution of all the students’ responses (0.512). On the contrary, these students did not perceive correctly the sampling variability, which was overestimated by 50% of students in 2nd grade and 38% of students in the 4th grade. We also observed that 21 students provided four identical values, therefore denoting a deterministic conception of sampling. Results in the older students (4th grade) were better both in the perception of expected value as well as in the perception of variability.

These results suggest the need to improve the teaching of sampling and to provide students with some experience of the random variability within different samples of the same population. In this sense, the use of some simulation applets which are freely available on Internet can help the teacher to make students’ conscious of sampling properties.

Keywords:

Sampling, sample proportion, variability and expected value, understanding, secondary school.