Introduction to Statistics: Mean, Median, Mode

Abstract

The goal of this lesson is to introduce the concepts of mean, median and mode and to develop understanding and familiarity with these ideas. The Measures Activity lets students explore mean and median in an efficient way; the Mean, Median and Mode Discussion helps them to formalize their knowledge.

Objectives

Upon completion of this lesson, students will:

  • understand three different measures of "center"
  • have been exposed to multiple ways of expressing a set of numbers
  • have practiced their arithmetic skills

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.

Sixth Grade

  • Statistics and Probability

    • Develop understanding of statistical variability.
    • Summarize and describe distributions.

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

    • 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
    • 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.

3rd Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of organizing, interpreting, analyzing and making predictions about data, the benefits of multiple representations of a data set, and the basic concepts of probability.

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

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

8th Grade

  • Patterns, Functions, and Algebra

    • 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 One: Looking at Data
    • Investigation Five: What Do We Mean by Mean?

6th

  • Module 4 - Statistical Safari

    • Section 2: Line Plots and Averages
      • Reason for Alignment: Mean, Median and Mode discusses the idea of "average". This is similar to how the terms mean, median, and mode are introduced in this section in the text. This lesson also gives additional practice problems and involves the activity "Measures".

Student Prerequisites

  • Arithmetic: Students must be able to:
    • sums, differences, and quotients for all activities.
  • Technological: Students must be able to:
    • Each student or group of students working together will need a computer with a web browser. Students should be comfortable using the computer and browser. Calculators may be helpful for solving problems that arise in discussions.

Teacher Preparation

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

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 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:

    • Does anyone know what "average" means?

  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 mean, median, and mode.
    • We are going to use the computers to learn about mean,median, and mode, 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

    • Lead a discussion, or the instructor can prepare a "live" discussion, to deepen and formalize the students' intuitive understanding of mean, median, and mode. (10-20 min)

  4. Guided Practice

    • Introduce and develop the concepts of mean and median with the Measures activity. Students will change parameters and discover patterns related to mean and median. Students can choose their own focus of measure, their own quantity, and their own units. (20 min)

  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

  • Combine this lesson with the Bell Curve Lesson for a look at how means are tied to distributions

Suggested Follow-Up

This lesson introduced the students to some basic ways of describing sets of data. The next lesson, Histograms and Bar Graphs, introduces histograms, bar graphs, and the concept of class interval. Students will learn to distinguish between bar graphs and histograms and to use each in the appropriate situations.