Univariate and Bivariate Data

Abstract

The following lesson is designed to introduce students to the differentiation between univariate and bivariate data. Students will gain experience determining what types of graphs and measures are appropriate for each type of data.

This lesson is designed for students who are familiar with graphs and measures related to univariate data, even if they don't know the vocabulary term.

This lesson is designed for students who are familiar with Shodor applets, particularly Bar Graph, Box Plot, Circle Graph, and Histogram.

Objectives

Upon completion of this lesson, students will:

  • be able to differentiate between univariate and bivariate data.
  • understand the different between categorical and numerical data.
  • understand what type of measure and representations to use for different data types.

Standards Addressed

Grade 6

  • Statistics and Probability

    • The student demonstrates an ability to classify and organize data.
    • The student demonstrates an ability to analyze data (comparing, explaining, interpreting, evaluating; drawing or justifying conclusions).

Grade 7

  • Statistics and Probability

    • The student demonstrates an ability to classify and organize data.
    • The student demonstrates an ability to analyze data (comparing, explaining, interpreting, evaluating, making predictions; drawing or justifying conclusions).

Grade 8

  • Statistics and Probability

    • The student demonstrates an ability to classify and organize data.
    • The student demonstrates an ability to analyze data (comparing, explaining, interpreting, evaluating, making predictions, describing trends; drawing, formulating, or justifying conclusions).

Grade 9

  • Statistics and Probability

    • The student demonstrates an ability to classify and organize data.
    • The student demonstrates an ability to analyze data (comparing, explaining, interpreting, evaluating, making predictions, describing trends; drawing, formulating, or justifying conclusions).

Grade 10

  • Statistics and Probability

    • The student demonstrates an ability to classify and organize data.
    • The student demonstrates an ability to analyze data (comparing, explaining, interpreting, evaluating, making predictions, describing trends; drawing, formulating, or justifying conclusions).

Grade 7

  • Statistics, Data Analysis, and Probability

    • 1.0 Students collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program

Grades 8-12

  • AP Probability and Statistics

    • 14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

Eighth Grade

  • Statistics and Probability

    • Investigate patterns of association in bivariate data.

Statistics and Probability

  • Interpreting Categorical and Quantitative Data

    • Summarize, represent, and interpret data on a single count or measurement variable
    • Summarize, represent, and interpret data on two categorical and quantitative variables
    • Interpret linear models

Grades 6-8

  • Data Analysis and Probability

    • Select and use appropriate statistical methods to analyze data

Grades 9-12

  • Data Analysis and Probability

    • Select and use appropriate statistical methods to analyze data

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 4: The learner will understand and use graphs and data analysis.

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 4: The learner will understand and use graphs and data analysis.

Introductory Mathematics

  • Data Analysis and Probability

    • COMPETENCY GOAL 3: The learner will understand and use graphs and data analysis.

7th Grade

  • Data Analysis & Probability

    • The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

8th Grade

  • Data Analysis & Probability

    • The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

Algebra II

  • Data Analysis and Probability

    • The student will collect, organize, represent, and interpret data; make and evaluate inferences and predictions; present and evaluate arguments based on data analysis; and model situations to determine theoretical and experimental probabilities.

Grade 8

  • Probability and Statistics

    • 12. The student uses statistical procedures to describe data.

7th Grade

  • Probability and Statistics

    • 7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including frequency distributions; line plots; histograms; stem-and-leaf plots; box-and-whisker plots; and scattergrams.

Secondary

  • Probability and Statistics

    • PS.01 The student will analyze graphical displays of data, including dotplots, stemplots, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers. Appropriate technology will be used to create graphical displays.
    • PS.02 The student will analyze numerical characteristics of univariate data sets to describe patterns and departure from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers. Appropriate technology will be used to calculate statistics.

Textbooks Aligned

Grade Six

  • Data About Us

    • Investigation Two: Types of Data

Grade Seven

  • Data Around Us

    • Investigation One: Interpreting Disaster Reports

Book 3

  • Module 8 - Making an Impact

    • Section 1: Collecting Data

8th

  • Module 1 - Amazing Feats and Facts and Fiction

    • Section 2: Displaying Data

Grade 8

  • Insights into Data

    • Representation of Data

Student Prerequisites

  • Mathematical: Students must be able to:
    • collect and organize data.
    • solve for statistical measures such as mean, median, and mode.
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Students will need

  • Access to a browser
  • Paper and pencil
  • Copy of worksheet

Teacher will need

Key Terms

average

It is better to avoid this sometimes vague term. It usually refers to the (arithmetic) mean, but it can also signify the median, the mode, the geometric mean, and weighted mean, among other things

bar graph

A diagram showing a system of connections or interrelations between two or more things by using bars

mean

The sum of a list of numbers, divided by the total number of numbers in the list. Also called arithmetic mean

median

"Middle value" of a list. The smallest number such that at least half the numbers in the list are no greater than it. If the list has an odd number of entries, the median is the middle entry in the list after sorting the list into increasing order. If the list has an even number of entries, the median is equal to the sum of the two middle (after sorting) numbers divided by two. The median can be estimated from a histogram by finding the smallest number such that the area under the histogram to the left of that number is 50%

mode

For lists, the mode is the most common (frequent) value. A list can have more than one mode. For histograms, a mode is a relative maximum ("bump"). A data set has no mode when all the numbers appear in the data with the same frequency. A data set has multiple modes when two or more values appear with the same frequency.

scatter plot

A graphical representation of the distribution of two random variables as a set of points whose coordinates represent their observed paired values.

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons and have students review:

    • Mean
    • Median
    • Mode
    • Range
    • Box Plots
    • Bar Graphs
    • Pie Graphs
    • Stem and Leaf Plots

    Ask students to brainstorm ideas about what univariate, bivariate, and multivariate data might be. Explain to students that they have already dealt with univariate data without using that vocabulary term.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today we are going to discover the differences between two types of data: univariate and bivariate.
    • Once we learn about the differences between these types of data, we will determine the appropriate graphs and measures for each type of data.
    • We will use the computers later, but please do not open them until I instruct you to do so.

  3. Teacher Input

    • Lead students in a discussion about the differences between univariate and bivariate data.
    • Lead students in a discussion about the differences between categorical and numerical data. Be sure to talk about how bivariate data can include both categorical and numerical data and that it can be represented using a multi-bar graph or scatter plot depending on the type of data.
    • Provide students with examples of how to display and analyze different types of data.
    • Introduce students to the Multi Bar Graph and Regression applets.

  4. Guided Practice

  5. Independent Practice

    • Have students work in pairs to collect data and complete a worksheet which asks them to create graphs of different types of data. Students can use the different applets listed above to create these graphs.

  6. Closure

    You may wish to bring the students back together to discuss what applets they choose to use to represent their data and any problems that we especially hard for the students.

Alternate Outline

This lesson can be rearranged if there is only one available computer:

  • While working on the worksheet, students can create data representations by hand and they can choose one graph to create using the appropriate applet and then share the graph with the class.

Suggested Follow-Up

This lesson could be followed by a series of discussion that explore bivariate data in more depth: