Box Plots

Abstract

The goal of this lesson is to introduce box plots and quartiles. An activity and discussion with supplemental exercises help students learn how data can be graphically represented.

Objectives

Upon completion of this lesson, students will:

  • have reviewed the concept of median
  • have learned how to calculate quartiles for any size data set
  • have learned how to build a box plot

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).

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

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.

7th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples.

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.

8th Grade

  • 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

7th

  • Module 5 - Recreation

    • Section 3: Proportions and Plots
      • Reason for Alignment: This lesson is at the introductory level. The discussions in the lesson explain the plots and how to find and interpret the various points of data. This lesson could be a good source for students, or parents, needing more explanation of the box plot. This type of data display is called a "Box-and-Whisker Plot" in the MATHThematics materials.

  • Module 7 - MATH-Thematical Mix

    • Section 4: Box-and-Whisker Plots and Circle Graphs
      • Reason for Alignment: This lesson revisits the Box Plot and has students work to understand the calculations and components of the plot.

8th

  • Module 1 - Amazing Feats and Facts and Fiction

    • Section 2: Displaying Data
      • Reason for Alignment: This lesson is a good review of key terms and encompases a thorough discussion of box plots. The more formal name for these box plots are called Box-and-Whisker Plots.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • put numbers in order from smallest to largest
    • calculate the average of two numbers
  • 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

  • Access to a browser
  • pencil and paper
  • The worksheet for the box plot activity's built in data sets
  • The worksheet for working with the data collected from the class (see below)

Key Terms

arithmetic mean

See mean

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

box plot

Also called box-and-whisker plot, this graph shows the distribution of data by dividing the data into four groups with the same number of data points in each group. The box contains the middle 50% of the data points and each of the two whiskers contain 25% of the data points.

histogram

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

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.

multimodal distribution

A distribution with more than one mode. The histogram of a multimodal distribution has more than one "bump"

range

The range of a set of numbers is the largest value in the set minus the smallest value in the set. Note that the range is a single number, not many numbers

total

A total is determining the overall sum of numbers or a quantity.

Lesson Outline

  1. Focus and Review

    Remind students what has been 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 examples of different ways that we have found to portray data?

  2. Objectives

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

    • Today, class, we are going to learn about box plots.
    • We are going to use the computers to learn about box plots, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

    • Remind students of the ideas behind means and medians, as covered in the Mean, Median and Mode discussion
    • Walk students through the construction of quartiles, the five-number summary and box plot construction as in the Box Plot discussion

  4. Guided Practice

    • Have students experiment with the built-in data sets available in the Box Plot activity to be sure that they understand how to read the the box plots. Questions for the data sets can be found in the worksheet.

  5. Independent Practice

    • Have the students collect the following data from each other:
      • Gender
      • Height
      • Length of ride/walk to school in minutes
      • Estimate number of hours of TV watched in a week.
    • Have the students explore the questions on box plots by building the appropriate box plots either by hand or using the Box Plot activity. With less mature students, it would be best to help them decide which box plot to graph for each question

  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.

  • Students can be asked to work with the notion of outliers, as can be done with the Box Plot activity

Suggested Follow-Up

If the students have not yet seen histograms, the lesson on Histograms and Bar Graphs makes a good follow-up. For more advanced students, The Bell Curve, covers the normal distribution and the bell curve controversy.