## Comparing Categorical and Numerical Data

3-5
1

In the final lesson of this unit, students recognize differences in representing and analyzing categorical and numerical data. Students also identify examples of each type of data.

In order to help students focus on the differences between categorical and numerical data, ask them to think about possible responses to the following questions and to decide which responses will provide categorical data and which will provide numerical data:

• How many pets do students in our class have?
• How many hours a week do we spend watching TV?
• What is our favorite sport?
• What kind of music do we like best?
• How many hours a week do we talk on the phone?
• What kinds of snacks do we like?
• How much do our backpacks weigh?
• How much candy do we eat each week?

For this activity students will need to be able to refer to two graphs they created in previous lessons:

Ask students what they notice about the appearance of the two graphs. They may point out that both are bar graphs (or line graphs). They may also notice that for the graph on students' heights, there are numbers along both axes; whereas on the graph about favorite books (leisure, or TV), there are words along one axis.

Ask students how the data are ordered on each graph. Numerical data are ordered numerically, but categorical data are ordered arbitrarily. Unlike the order of numerical data, the order of categorical data can be changed without affecting the analysis. Because there is no established order for the arrangement of categorical data, the shape of the data set will vary, depending on the order. Therefore, the shape of the data is not used in the analysis of categorical data.

Talk with students about how the data sets are analyzed. Since students have just finished working with numerical data, it is likely that they will mention the median, range, and outliers. Help students realize that these measures are used for the analysis of numerical data but not categorical data. Have them explain why this is so. (The order of the data can be changed since it is not numerically ordered.) The mode, on the other hand, does not depend on the order of the data, but on the frequency, so it can be identified in both categorical and numerical data. Fractional or proportional relationships can also exist within both types of data. Throughout this discussion, it is important that students provide specific examples by referring to the graphs they created in previous lessons. The students might conclude that mathematically, there are more ways of talking about numerical data than about categorical data.

Assessment Option

Ask students to create or find some categorical data, and represent it with a bar graph.
Ask students to go to create or find numerical data, and then create a graph that represents that data. (For example, the graph below shows some imaginary data for heights of boys in some classroom.)

With partners, students should analyze both graphs. Discuss the observations in class. Then summarize the differences/similarities in the analyses for each kind of data.

Teacher Reflection

• Did students achieve the objectives for this lesson? What evidence supports this claim? What changes should I make to create a more effective lesson?
• What additional experiences do students need to be successful with this activity?
• Were students able to explain their reasoning in a clear and logical manner?
• What additional extensions/experiences would be appropriate?

### Categorical Data

3-5
In the first lesson of this unit, students formulate and refine questions that can be addressed with categorical data. They consider aspects of data collection such as how to word questions and how to record the data they collect. Finally they represent and analyze the data in order to answer the question posed.

### Numerical Data

3-5
In the second lesson of this unit, students pose and refine questions that can be addressed with numerical data. They consider aspects of data collection such as how to obtain measurements and record the data they collect. They represent and analyze the ordered numerical data by describing the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed. In collecting data, students measure with standard units and carry out simple unit conversions, such as from centimeters to meters or feet to inches.

### Learning Objectives

Students will:

• Identify examples of categorical and numerical data.
• Recognize differences in representing and analyzing categorical and numerical data.

### NCTM Standards and Expectations

• Compare different representations of the same data and evaluate how well each representation shows important aspects of the data.

### Common Core State Standards – Mathematics

Grade 3, Num & Ops Base Ten

• CCSS.Math.Content.3.NBT.A.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.B.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP6
Attend to precision.