Histograms and Bar Graphs

Abstract

The goal of this lesson is to introduce histograms, bar graphs and the concept of class interval. An activity and three discussions with supplemental exercises help students learn how data can be graphically represented (and mis-represented). Students will learn to distinguish between bar graphs and histograms and to use each in the appropriate situations.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to histograms and bar graphs
  • have understood the difference between bar graphs and histograms
  • have seen examples of how data can be represented in a variety of graphical forms

Standards Addressed

Grade 3

  • Statistics and Probability

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

Grade 4

  • 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 5

  • 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 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).

Third Grade

  • Measurement and Data

    • Represent and interpret data.

Fourth Grade

  • Measurement and Data

    • Represent and interpret data.

Fifth Grade

  • Measurement and Data

    • Represent and interpret data.

Statistics and Probability

  • Interpreting Categorical and Quantitative Data

    • Summarize, represent, and interpret data on a single count or measurement variable

Grades 6-8

  • Data Analysis and Probability

    • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
    • Select and use appropriate statistical methods to analyze data

Grades 9-12

  • Data Analysis and Probability

    • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

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

  • Algebra

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

  • Data Analysis and Probability

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

6th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of the relationships within one population or sample.

4th Grade

  • Data Analysis and Probability

    • Standard 4-6: The student will demonstrate through the mathematical processes an understanding of the impact of data-collection methods, the appropriate graph for categorical or numerical data, and the analysis of possible outcomes for a simple event.

3rd Grade

  • Probability and Statistics

    • 3.22 The student will read and interpret data represented in line plots, bar graphs, and picture graphs and write a sentence analyzing the data.

7th Grade

  • Probability and Statistics

    • 7.16 The student will create and solve problems involving the measures of central tendency (mean, median, mode) and the range of a set of data.
    • 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.

5th Grade

  • Probability and Statistics

    • 5.17c The student will create a problem statement involving probability and based on information from a given problem situation. Students will not be required to solve the created problem statement.

8th Grade

  • Computation and Estimation

    • 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data, such as finding a discount and discount prices and balancing a checkbook.
    • 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data,

  • Patterns, Functions, and Algebra

    • 8.17 The student will create and solve problems, using proportions, formulas, and functions.
    • 8.18 The student will use the following algebraic terms appropriately: domain, range, independent variable, and dependent variable.
    • 8.18 The student will use the following algebraic terms appropriately: domain, range,

  • Probability and Statistics

    • 8.12 The student will make comparisons, predictions, and inferences, using information displayed in frequency distributions; box-and-whisker plots; scattergrams; line, bar, circle, and picture graphs; and histograms.
    • 8.12 The student will make comparisons, predictions, and inferences, using information

Textbooks Aligned

Grade Six

  • Data About Us

    • Investigation Two: Types of Data

7th

  • Module 5 - Recreation

    • Section 2: Surveys and Data Displays
      • Reason for Alignment: This lesson shows uses for each, and a discussion of when to use each type of graph. This should be easily understood by students at this level. There are data sets included for practice and analysis.

8th

  • Module 1 - Amazing Feats and Facts and Fiction

    • Section 1: Rates and Data Displays
      • Reason for Alignment: This lesson should help students sort out the differences between bar graphs and histograms and help them understand their specialized uses. There is a good discussion suggested for this. This lesson should also help students as they use their own data for each of these types of data displays.

Student Prerequisites

  • Geometric: Students must be able to:
    • recognize how data can be plotted and interpreted from shapes such as bar graphs
  • 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

Key Terms

bar graph

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

class interval

In plotting a histogram, one starts by dividing the range of all values into non-overlapping intervals, called class intervals, in such a way that every piece of data is contained in some class interval

histogram

A bar graph such that the area over each class interval is proportional to the relative frequency of data within this interval

Lesson Outline

  1. Focus and Review

    Remind students of what they have learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • What are some different ways to represent data?
    • Is there one way that is superior to all the others?

  2. Objectives

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

    • Today, class, we are going to learn about the differences between bar graphs and histograms.
    • We are going to use the computers to learn about the differences between bar graphs and histograms, but please do not turn your computers on until I ask you to. I want to show you a little about these activities first.

  3. Teacher Input

    • Use the Bar Graph activity to familiarize students with visually representing data. Be sure to demonstrate the importance of vertical scale in bar graphs.
    • Use the Histogram activity to demonstrate the importance of class interval size and vertical scale in histograms.

  4. Guided Practice

    • Lead a class discussion about Class Intervals to deepen and formalize students' intuitive understanding of histograms, bar graphs, and class intervals. Use Sample Problems on Bar Graph Scales to provide examples for the discussion. Students can read the sample problems and try to solve them individually or in groups, or use them as examples and create their own statistics problems for themselves and/or for other students.
    • Next, initiate a discussion based on Vertical Scale to enhance the students' understanding of how data can be represented graphically.
    • To bring together ideas introduced in the lesson so far, the Histogram vs. Bar Graphs Discussion will help students to compare the properties of histograms and bar graphs.

  5. Independent Practice

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways.

  • Instead of initiating class discussions, have the students read the discussions independently, and then see how well they understood by having them complete the sample problems worksheets that accompany each discussion.
  • For further investigation into the representation of data, and to practice working with fractions and decimals, students can work with Sample Problems on Data Abuse.

Suggested Follow-Up

After the discussions and activities, the students will have seen examples of bar graphs and histograms, and be able to tell them apart and know the circumstances when each should be used. The next lesson, The Bell Curve, covers the normal distribution and the bell curve controversy.