D1.3 Select from among a variety of graphs, including stacked-bar graphs, the type of graph best suited to represent various sets of data; display the data in the graphs with proper sources, titles, and labels, and appropriate scales; and justify their choice of graphs.
Skill: Choosing the Appropriate Graph to Represent Various Sets of Data
Visualization involves representing data in graphs and tables, helping in the effective communication of information for the purpose of interpretation.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 62.
How to Organize the Data
A variety of tables and graphs are commonly used in data management to represent sets of data clearly and to facilitate analysis. Each has its advantages as well as its limitations. It is important to note, however, that the construction of a table generally precedes the construction of a graph.
Students in Grade 5 represent data using a variety of graphs, including multiple-bar graphs and stacked-bar graphs.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 66-67.
Skill: Representing Data Sets
Representation of data in tables and graphs is used to communicate information for interpretation.
Students must determine the best way to group the data collected in order to facilitate their analysis, and then choose an appropriate representation and construct it in such a way as to ensure the message is conveyed with clarity.
Once the data has been collected and recorded, students must group it within a limited number of categories. There is no rule that dictates how to group the data. The choice of grouping depends largely on the type of data collected.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 80-81.
Once the data from an inquiry have been grouped into categories in a frequency table or similar, it is often very useful to represent them using a graph because of its visual impact. In effect, a graph:
- presents information in an organized manner;
- is generally easier to read and interpret than data represented using text or a table;
- allows you to see the set of data at a glance, getting a first impression of it; for example, whether the data is evenly or unevenly distributed among the categories;
- facilitates the next step in the inquiry process, which is the analysis of the data and interpretation of the results.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 86.
For example, students can represent a set of statistical data much more succinctly and effectively with a table or graph than with an oral description. However, they must learn to use the various conventional representations of data (for example, frequency table, pictograph, bar graph) correctly to ensure that the message being conveyed is accurate and that it is understood by others. Students can then use these representations to justify a reasoning or support an argument during a mathematical discussion with the class.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 30.
The source, titles, labels, and scales provide important information about data in a graph or table:
- The source indicates where the data was collected.
- The title introduces the data contained in the graph.
- Labels on the axes of a graph describe what is being measured (the variable). A key on a stacked-bar graph indicates what each portion of the bar represents.
- Scales are indicated on the axis of bar graphs , showing frequencies , and in the key of pictographs.
- The values for relative frequencies are indicated using fractions, decimals, or percents.
The order of the categories is important in graphs with quantitative data; numbers are put in ascending order. In contrast, the order of the categories is not important in graphs with qualitative data; for example, the colours can be put in any order.
Skill: Justifying Choice of Graph
In order to select the best representation of the data collected, students need to understand the characteristics and purposes of different types of graphs. While data can be visualized using different types of representations, it is important to select the one that provides the best clarity and accuracy of the data in context. Pictographs, tally tables, bar graphs, and multiple-bar graphs, for example, are used to present both qualitative data and discrete quantitative data.
Students in Grade 5 are representing sets of data for the first time in stacked-bar graphs; however, it is important to remind them of the types of graphs seen in previous grades so that they can choose the appropriate graph for a given data set and justify their choice.
Knowledge: Types of Graphs
Line Plot
The line plot is useful for representing a large amount of numerical data. Each piece of data is represented by an X. Xs that correspond to the same quantity are arranged one above the other to form columns. Since the height of each column corresponds to the frequency it represents, data analysis is made easier.
Example
The teacher asks students to check the number of years their parents have lived in the current home. The results are represented by the following line plot.
image Number Line. The title is: Number of Years Parents of Grade 4 Students Have Been Living in Current Home The subtitle is: Number of years. It is graduated from 1 to 23. 1 year is 1 x, 3 years is 4 x, 4 years is 6 x, 5 years is 5 x, 8 years is 2 x, 12 years is 1 x, 13 years is 7 x, 14 years is 4 x, 15 years is 5 x, 20 years is 2 x and 23 years is 1 x. For the other years, it is zero x.
It is easy to see from the distribution of X's that there were two waves of moves, 3-5 years ago and 13-15 years ago, which might be because of new housing construction or the creation of new jobs in the city.
Characteristics of a line plot:
- it has a title;
- It has a graduated horizontal axis with a label.
- each data is represented by an X. The Xs that represent the same numerical data are aligned and are separated from each other by a constant space.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 83-84.
Bar Graph (Vertical or Horizontal)
The bar graph is used to represent the frequencies of a data set. It is composed of rectangular bars whose length corresponds to the size of the frequencies. It allows the reader to see, at a glance, the distribution of the data in each category.
Note: To facilitate the construction of bar graphs, it is recommended that students be instructed to use graph paper.
image Graph1: The title of the vertical bar graph is: Birthdays according to seasons. The title of the x-axis is: Seasons. The x-axis is graduated in 4 seasons: Spring, Summer, Fall and Winter. The title of the y-axis is: Number of students. The y-axis is graduated by one, from 0 to 9. graph2: The title of the horizontal bar diagram is: Birthdays according to seasons.The title of the x-axis is: Number of students. The x axis is graduated by one, from 0 to 9. The title of the y-axis is: Seasons. The y-axis is graduated in 4 seasons: spring, summer, fall and winter. Graph2: The title of the horizontal bar diagram is: Birthdays according to seasons.The title of the x-axis is: Number of students. The x axis is graduated by one, from 0 to 9. The title of the y-axis is: Seasons. The y-axis is graduated in 4 seasons: spring, summer, fall and winter.
Characteristics of a bar graph:
- It has a title (for example, Birthdays by season).
- It has an axis (vertical or horizontal) graduated on an appropriate scale.
- It has another axis that represents categories (for example, spring, summer, fall, winter).
- The two axes each have a label (for example, Number of Students, Seasons).
- The bars have the same width and their length corresponds, depending on the scale chosen, to the size of the frequency they represent (for example, if the scale is by intervals of 1, a bar of length 3 represents a frequency of 3).
- The bars are separated by equal spaces.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 71-72.
Multiple-Bar Graph
The multiple-bar graph represents more than one set of data simultaneously to facilitate comparisons between them. It has the same characteristics as a bar graph, but each category has two or more bars of data. A key indicates what aspects are represented.
A multiple-bar graph can be constructed in either a horizontal or vertical orientation. The orientation chosen could depend on personal preference or on the purpose of the representation.
The title of the Vertical Multiple Bar diagram is: Rainwater collected in April. The title of the x-axis is: Day of the week. The axis is graduated in four weeks, i.e. week 1, week 2, week 3 and week 4. The title of the y-axis is: Rainwater quantity measured (in millimeters). The axis is graduated by 5, from 0 to 25. There is a legend: blue is Monday, yellow is Tuesday, red is Wednesday, light blue is Thursday and green is Friday. Below the legend it says "Source: School B Grade 5 Students".
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
The data can be organized in a way that makes it easier to compare categories of data with each other. For example, to compare the number and type of pets of students in Grades 1-6, data for Grades 1 to 3 and Grades 4 to 6 could be combined and reprensented in a double bar.
image The title of the horizontal multi bar diagram is: Pets of students in grades 1 to 6. The title of the x-axis is: Number of Students. The axis is graduated by 10, from 0 to 50. The title of the y-axis is: Pets. The axis is scaled by category, namely, "no animal", "other animal", "fish", "bird", "cat", and "dog". Below the x-axis is a legend: yellow, Students from grades 1 to 3; purple, Students from grades 4 to 6.
Stacked-Bar Graph
Stacked-bar graphs present data values proportionally. These graphs can be used to present percentages, or relative frequencies. Each bar in the graph represents the whole, and each of the segments in a bar represents a part (a different category). Different colours are used within each bar to easily distinguish between categories.
Stacked bars are useful when there is a second categorical variable. The first (and primary) variable is shown along the entire length of the bar, and the second variable is represented as stacks within each bar.
The source, titles, labels, and scales provide important information about data in a graph or table:
- The source indicates where the data was collected.
- The title introduces the data contained in the graph.
- Labels on the axes of a graph describe what is being measured (the variable). A key on a stacked-bar graph indicates what each portion of the bar represents.
- Scales are indicated on the axes of bar graphs, showing frequencies , and in the key of pictographs.
- The values for relative frequencies are indicated using fractions, decimals, or percents.
Example
image The title of the stacked bar diagram is: Rainwater collected in April. The title of the x-axis is: Day of the week. The axis is scaled into four weeks: week 1, week 2, week 3 and week 4. The title of the y-axis is: Rainwater quantity measured (in millimeters). The axis is graduated by 5, from 0 to 60. There is a legend: red is Monday, yellow is Tuesday, pink is Wednesday, green is Thursday and blue is Friday. Under the legend it says "Source: School B Grade 5 Students".
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.