With the arrival of big data era, substantial changes have taken place in different fields of society, such as the emergence of artificial intelligence, machine learning, precision medicine, and computational education (François and Monteiro, 2018). Data has become an essential asset in various professions for daily use, and data analysis ability correspondingly becomes a necessity for our work and life (Dani and Joan, 2004). In fields such as business, economics, and others, they increasingly rely on data analytics rather than experience or intuition only to make decisions in management (Bryant et al., 2008). Due to the popularity of data in daily life and workplace, data literacy has gradually become an ability that everyone shall have instead of an ability mastered by only few senior personnel in some specific industries (Borges-Rey, 2017; Sharma, 2017). Prado and Marzal (2013) have provided a list of standards for data literacy, which include (1) Determining the context of data production and reuse, as well as the value, category and format of data; (2) Figuring out when you need data and obtain it appropriately; (3) Properly evaluating data and data sources; (4) Using certain plans, measures, system architecture and appropriate evaluation methods to determine the appropriate methods to operate and analyze the data; (5) Visualization of data analysis results; (6) Using the analysis results to learn, make decisions, or solve certain problems. This list covers almost all required in data literacy and has become one of the basic qualities necessary for the future use. Moreover, data literacy also plays a crucial role in mathematics. Generation and solution of some certain mathematical problems are based on data analysis as well. Good data analysis ability can effectively help students find and solve problems. Therefore, it is of far-reaching significance to focus on cultivating students’ data analysis ability.

Commonly cultivated by means of statistical disciplines or statistical content, data analysis ability tightly relates with statistics and can be extracted from statistical literacy. However, statistical literacy so far has not been well defined and there is no consensus achieved on its definition among statistics educators, statisticians, and researchers globally (Kaplan and Thorpe, 2010; Schield, 2010; Ridgway et al., 2011; English, 2014). Moore and Cobb (2000) took “change” as the core element of statistical thought and believed that “change” was everywhere in the process. Wallman (1993) stated that statistical literacy was the ability to understand and critically evaluate the statistical results that permeate our daily lives, as well as the ability to understand the contributions of statistical thinking in the public, professional, personal, and private spheres. Through the literature, we can see that there exist common core elements in both the statistical literacy and data analysis ability. Pedagogically, data analysis offers an opportunity for students to explore openly. Conceptually, data analysis examines patterns, hubs, clusters, gaps, dissemination, and variation in data. Philosophically, some advocates of data analysis recommend introducing data in a non-probabilistic environment, while others suggest establishing a link between data analysis and the notion of probability. In the latter view, both data and contingency were considered in the framework of a systematic study of probability (Shaughnessy et al., 1996). In this study, we extracted the data analysis ability from the statistical literacy as the evidence for its definition. Specifically, we defined data analysis ability as the ability to perceive data from daily life, to consciously collect and organize data, to represent data according to different needs, and to rationally analyze and interpret data through operations.

In the assessment of data analysis ability, Graham et al. (2009) built a framework representing children’s statistical thinking based on the Cognitive Diagnostic Model (CDM) and other relevant research. This framework provided the theory for the characterization of children’s statistical thinking and planning guidance for data processing. Reading (2002) established a four-level statistical thinking analysis framework, which were Data Collection, Data Tabulation and Representation, Data Reduction, and Interpretation and Inference. Using the Structure of the Observed Learning Outcome (SOLO, Biggs and Collis, 2014) classification framework, he analyzed the typical responses of students in different levels and drew corresponding conclusions. Mooney (2002) portrayed statistical thinking into four dimensions: Describing data, Organizing and Generalizing data, Representing data, and Analyzing and Interpreting data. Still based on the SOLO classification framework, four levels were described for each aspect: Idiosyncratic, Transitional, Quantitative, and Analytical. What’s more, they also pointed out the necessity of establishing learning trajectories that connect students with different levels of statistical thinking. Due to the high expectation in statistics teaching (Jacobe et al., 2014), effective assessment tools were in a great necessity to evaluate learners’ understanding of statistical concepts more precisely.

As we can see, people have sufficiently realized the significance of data analysis and put the cultivation of students’ data analysis ability in a more centered position. Although there is no clear definition of data analysis ability yet, a lot of research has been conducted on its attribute division, which lays the foundation for future research. Currently, the assessment of students’ data analysis ability has mostly constructed the students’ levels of abilities according to the traditional Classical Testing Theory (CTT). It exhibits great advantages in understanding the overall status of students’ data analysis ability, however, it cannot provide students with more fine-grained diagnostic information, which is the key to promoting students’ development (Huff and Goodman, 2007). The new Cognitive Diagnostic Assessment (CDA), a measurement tool developed in the most recent years combining both cognitive psychology and modern psychometrics, can analyze students’ knowledge or skill attributes involved in the process of answering questions from the perspective of cognitive psychology, and integrate attributes into the measurement model. The individual’s psychological cognitive process is measured to determine students’ mastery of attributes. CDA provides students with more detailed diagnostic information and supports a further in-depth study of students’ cognitive process (Leighton and Gierl, 2007). In this study, CDA was applied to construct a cognitive model of students’ data analysis ability. Through an examination of 503 Grade 9 students in China, the probability of mastering the attributes of students’ data analysis ability can be obtained. On this basis, the students’ learning path, learning progression, and personal assessment were then explored, providing a reference for the further study of students’ data analysis ability.

In the attribute classification of data analysis ability, data processing includes organization, description, presentation, and analysis of data, which strongly depends on the representation of various graphics and icons (Shaughnessy et al., 1996). Jones et al. (2000) formed a model composed four attributes of data analysis ability, namely, Data organization, Data representation, Data analysis, and Data interpretation. In this model, the continual process of data analysis was systematically presented. What’s more, they also made detailed operational definitions for each attribute and coded them accordingly. Mooney (2002) portrayed statistical thinking as four dimensions: Data description, Data organization and generalization, Data representation, Data analysis, and interpretation. Reading (2002) concluded a few steps in general, which included Data collection, Data tabulation and presentation, Data summarization, interpretation, and inference in the established statistical thinking framework. Arican and Kuzu (2020) divided data analysis literacy into four aspects: (a) Data representation and interpretation, (b) Sample interpretation, (c) Statistical methods selection, (d) Understanding and application. These studies provided the basis for the classification of data analysis ability attributes.

The construction of the cognitive model, especially the construction of the relationship between the attributes in the cognitive model, is more complex. Normally, expert method is used to analyze the cognitive process of students in a certain field, to obtain the relationship between different attributes. In this study, five experts were selected, including three middle school math teachers who were awarded the title of Guizhou Provincial or Municipal Famous Teachers, with rich teaching experience and knowledge of students. Another expert was a mathematics education researcher who focuses on the assessment of data analysis ability and can inspect the cognitive model of data analysis ability from the theoretical level. The last expert we invited was a doctoral student majoring in mathematics education, who has been engaging in the teaching of data problem-solving for a long time, and can examine the cognitive model of data analysis ability from the perspective of assessment.

Through open questionnaires, experts were required to enumerate the attributes of data analysis ability and draw the structural relationships between the attributes. Results of the expert method were presented in Table 1.

Based on the common elements in Table 1 and the existing definitions of data analysis ability, the attributes of data analysis ability were extracted, which were: (1) Data awareness; (2) Data collection and sorting; (3) Data representation; (4) Data concentration; (5) Dispersion of Data; (6) Data interpretation and reasoning. According to the relationship between the extracted elements in Table 1, the structure model of data analysis ability was obtained in Figure 1.

The subjects of this study were 503 Grade 9 students from 10 classes in two middle schools in Guizhou Province, China. Guizhou is a relatively underdeveloped province in terms of education. Based on the annual proportion of admission to the secondary school entrance examination, the two schools selected belong to the upper middle level in Guizhou Province. The test was 1 h long and all Grade 9 students in both schools took the test. Informed consent was obtained from all the students and teachers before the implementation of test.

The probability of each student’s attribute mastery can be obtained through the assessment of the model in CDA. To further analyze the gender differences in students’ mastery of attributes, comparison analyses of the attribute mastery probability were conducted on genders and the mastery probabilities of the six attributes were obtained as shown in Figure 2.

Figure 2 showed that the mastery probability of the attribute Know data was the highest, reaching 100%, which indicated that almost all students had the basic data awareness. The probability of mastering these four attributes, Data collection and collation, Data presentation, Concentration of data, and Dispersion of the data, were quite similar at about 70%. Among these four, the probabilities of mastering attributes of Data collection and collation and Dispersion of the data were slightly higher. Last, the mastery probability of Interpretation and inference of data was the lowest, about 60%.

Overall, in Figure 2, what we can observe was that there was no obvious gender difference in attribute mastery probability, especially the probability of mastering Know data, Data presentation and Dispersion of data were almost the same for both male and female students. In terms of Data collection and collation, Concentration of data and Interpretation and inference of data, the probabilities of male students mastering attributes were slightly higher than these for female ones, showing certain advantages for male students.

With a combined method of learning progression construction with Item Response Theory (IRT), students’ abilities were calculated in each type of knowledge state, and students in different knowledge state categories were divided (Wu et al., 2021). The ability value 1.37 was the highest for the knowledge state (111111), and the ability value −2.41 was the lowest for the knowledge state (010000). We have divided the interval (−2.5, 1.5) into 5 levels and the diagram for learning progression was shown in Figure 3.

According to the attributes of the knowledge states in the different stages in Figure 3, the learning progression of the ladder were defined, and the divisions of learning progression level were organized in Table 7. It provided a more reliable theoretical basis for student learning, teacher teaching, textbook compilation, and test compilation.

Accurately depicting the knowledge state of each student is the greatest advantage of CDA. To fully illustrate the fine-grained information that CDA can provide for each student, three students numbered 337, 476 and 424 were selected for the analysis. The radar chart of their attribute mastery was shown in Figure 4.

The characteristic of the three students shown in Figure 4 was that they correctly answered the same number of questions. If through the traditional evaluation methods, these students were considered to have the same total score, showing that there was no difference among them. However, in CDA, differences were still obvious in Figure 3. Student No. 337 had mastered attributes ①, ②, ④, ⑥; student No. 476 had mastered attributes ①, ②, ③, ⑤, ⑥, but not fully mastered attributes ② or ⑥, whose probability of attribute mastery was approximately 80%; and student No. 424 had mastered the attributes ①, ②, ④, ⑤, ⑥. Therefore, what we can conclude was that these three students not only differed in the type of attribute mastery, but also in the number and degree of attribute mastery. These results offered more fine-grained information for students’ personalized learning.