# Errors concerning Statistics and Probability in Spanish Secondary School Textbooks
## 3 Results
### Errors Detected in Texts on Descriptive Statistics
#### 3.1.3 Uncritical Choice of Graphic Representation  
Errors can also be found in the type of graph used to represent the information. Thus, although many books use ordinal variables to exemplify the pie chart, it would be more appropriate to select a bar chart for this type of variable and to reserve the pie chart for nonordinal qualitative variables. None of the textbooks analysed provide any indications about the adequacy of one or another form of graphic representation according to the observed data, nor about the resulting legibility and interpretability. No warnings were given, either, about the difference between the use of bar charts for qualitative or for quantitative variables, or about what type of scale should be considered for the horizontal axis. Moreover, the textbook examples usually consider the application of both types of figure for both types of variables, and the exercises frequently require the student to obtain the two graphic representations for the same set of data. In addition to this problem regarding criteria for selecting the type of graphic representation, the frequency polygon is included in the curriculum for these courses. This concept is confused in all the texts, which provide a definition that corresponds to a polygonal line that joins the highest points of the bars from a bar chart. None of the texts provides an interpretation of this graph, and the polygon is never completed by closing the ends with segments that join the \(X\) axis with the first and last of the bars. Obviously, using a frequency polygon or part of it to represent a nonordinal qualitative variable--a common feature of these texts--is hard to interpret and contributes very little to understanding the question. Figure 1: Incorrect (**left**) and correct (**right**) representation of the statement given. 3.1.4. Measures, Estimators and Parameters  
The term \"parameter\" is sometimes used in reference to the arithmetic mean or other measures of position or dispersion. In this context, the word \"parameter\" is extremely inappropriate because the text does not deal with population data but with small data sets. The correct expression, then, would be \"measures\". The difference between these two terms is the fundamental basis of statistical inference and transcends this topic to become a core element of statistical versus mathematical thinking. Moreover, at higher levels, the difference between the two concepts becomes part of the content. Specifically, in the second year of high school (Grade 12), the curriculum subject of Mathematics Applied to Social Sciences II includes \"Parametric statistics: parameters of a population and statistics obtained from a sample\". In their analysis of university entrance exam performance in activities related to statistics, [PERSON] et al. [48] found that high school students (in the subject \"Mathematics Applied to Social Sciences\") had difficulty discerning between parameter, estimator, and estimation. The fact that students' textbooks label the basic measurements of position and dispersion as \"parameters\" may be a relevant factor in the failure to make this necessary distinction. 3.1.5. Discrete Measures for Continuous Variables and Vice Versa  
One error that is frequently observed is to consider a numerical variable, which could be taken as continuous, and then making discrete observations of it, thus generating intervals and diluting the discrete information into intervals for a continuous variable. Subsequently, the measures are calculated using the class mark, which rediscreises the values, but does so inaccurately. In this respect, consider the following example: \"A group of children take the following time (in minutes) to travel from their home to the school: 34, 12, 9, 21, 5, 2, 20, 18, 4, 9, 16, 18, 14, 7, 32, 10, 21, 6, 11, 13. Summarise the information in a table containing intervals of 5 minutes, starting at 0. With the table obtained, calculate the position measurements and dispersion\".