Graphical (visual) representation of statistics

As mentioned before, statistics and probability also involve visual representation through tables, charts, graphs, and diagrams. These tools can summarise large amounts of data and allow individuals to compare data, see relationships, and spot trends.

ACARA (n.d.), describes some aspects of graphical representation of statistics that students will learn within the Australian Curriculum.

These can be summarised as:

• Choosing the relevant graphical representation to show data
• Using features of graphical representations to make predictions
• Showing the continuous variables depicting change over time interpreting graphs depicting motion, patterns in real life situations, and outliers
• Determining whether data is from a sample or population and also what type of sample to use from a population
• Making reasonable statements about a population when graphically presenting some sample data

As part of statistics and probability, students learn to create graphical representations of data and to be able to interpret them. The curriculum in Australia includes various graphical charts and tables, including data tables, Venn diagrams, pie charts, histograms, scatter plots, and box and whisker plots. One challenge students may need to deal with in this regard is to negotiate and choose the best chart to represent the data as well as label and organise them appropriately. Such abilities can be viewed as higher order skills (Curcio, 1987) that need to be developed over time.

The below table by MooMooMath and Science presents the different types of graphs and charts and when to use them.

Note: there is a description of this image in the downloads section.

Research suggests that students have a number of misconceptions with graphs (Mevarech & Kramarsky, 1997). These misconceptions will be discussed in more detail in the next topic, but some of the common problems include:

• Difficulty between different types of presentation of charts
• Limited skills in extracting information from or explaining the contents of a chart
• An inability to apply learnings into other contexts, for instance in other subject areas

In constructing charts and graphs, Wainer (1992) proposes that three levels of information processing (described as ‘graphicacy’) are required to understand charts and graphs.

These are:

1. Elementary level: this level involves extraction of information from graphical data (e.g., identifying the highest score).
2. Intermediate level: this level involves seeing trends in data (e.g., increase or decrease in mean daily temperatures in summer months from 1970-2020).
3. Deep level: this level involves providing overall answers to questions (e.g., making predictions about uptake of fossil fuels in the 2030s in light of increasing use of green energy).

The first and second levels would likely be within the grasp of a statistically literate individual. However, the third one requires being able to construct or read sophisticated charts and make inferences from them, which will likely be beyond most people who do not work in the STEM area, statistics or data science. As a result, it should be noted that moving from an elementary to an intermediate level may be challenging for students and require skilled teaching.

One approach that can help in scaffolding understanding is the idea of working with students’ alternative conceptions. An alternative conception is another way of thinking why students misconceive certain mathematical phenomena (Fuji, 2014). Rather than simply misunderstanding a concept, an alternative concept is when the students bring their own worldview and experiences to make sense of a concept. Within a graphical representation of statistics, for example, students may colour in different bars on a histogram as they would for a bar chart or pie chart.

Their alternative conception is reasonable as they have seen other charts (even ones that look very similar) have multiple colours. However, there is an underlying misunderstanding of the difference between categorical (use a bar chart with different colours and spacing) and numerical data (use a histogram). One suggested approach to help address this problem is to allow students to draw graphs without initially using the terminology of bar chart and histogram and then discussing with them any mistakes made by focusing solely on the data (Whitaker & Jacobbe, 2017).

Australian Curriculum, Assessment and Reporting Authority (ACARA). (n.d.-d). National numeracy learning progression. Statistics and probability. https://www.australiancurriculum.edu.au/resources/national-literacy-and-numeracy-learning-progressions/national-numeracy-learning-progression/statistics-and-probability/?subElementId=50841&scaleId=0

Curcio, F. R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for research in mathematics education 18(5), 382-393. https://www-jstor-org.ezproxy.usq.edu.au/stable/749086

Fuji T. (2014) Misconceptions and Alternative Conceptions in Mathematics Education. In Lerman S. (Ed.) Encyclopedia of Mathematics Education. Springer. https://doi.org/10.1007/978-94-007-4978-8_114

Mevarech, Z. R., & Kramarsky, B. (1997). From verbal descriptions to graphic representations: Stability and change in students’ alternative conceptions. Educational Studies in Mathematics, 32(3), 229-263. https://doi.org/10.1023/A:1002965907987

MooMooMath and Science. (2018). Types of graphs and when to use them. MooMooMath and Science. Accessed from https://www.youtube.com/watch?v=yrTB5JSQPqY&t=53s.

Wainer, H. (1992). Understanding graphs and tables. Educational Researcher, 21(1), 14–23. https://journals-sagepub-com.ezproxy.usq.edu.au/doi/abs/10.3102/0013189×021001014#articleCitationDownloadContainer

Whitaker, D., & Jacobbe, T. (2017). Students’ understanding of bar graphs and histograms: results from the locus assessments. Journal of Statistics Education, 27(2). https://www-tandfonline-com.ezproxy.usq.edu.au/doi/full/10.1080/10691898.2017.1321974