SHOW TOCHIDE TOC

Interactivity

Interactivity


Introduction: Exploring Measures of Central Tendency

An average is a single number that characterizes a data set. Averages can be misleading, however, if you're not careful. Suppose the manager of a small store earns $95,000 per year, while the other five employees earn $15,000 to $30,000 each. Although the average amount earned per year is $35,000, that single amount doesn't represent yearly earnings very clearly.

Using the Interactivity, you will explore three measures of central tendency: the mean (or average), the median, and the mode. Each measure describes the central tendency of a data set in a different way. You will also explore the range, which is a measure of how spread out a data set is.

Exploring Measures of Central Tendency

Exploration Guide

In this Interactivity, you will be looking at the yearly salaries of people working at a particular company.

  1. Click "New" to generate a data set. Stacks of coins will appear on the graph. Each coin represents one person's annual salary, in thousands of dollars per year. For instance, if there were four coins in the "60" column, that would mean four people at the company earn $60,000 per year.
    1. What is the lowest salary at this company?
    2. What is the highest salary at this company?
    3. What is the most common salary?
    4. How would you describe this data set?
  1. The "mean" (or "average") of a data set is the sum of each of the data values divided by the number of data values in the set. In terms of this scenario, if you took all the salary money and divided it evenly among all the employees, the "mean" is what each employee would get.
    1. Turn the "Show mean" option on. A purple vertical line will appear showing the value of the mean.
    2. Try dragging some coins from one column to another to modify the data set.
    3. Does the mean change when you drag any of the coins on the graph to the left? Does it increase or decrease? What happens when you drag a coin to the right?
  1. The "median" of a data set is the "middle" value. If you took all the salaries of the employees and sorted them in order from lowest to highest, the middle salary would be the median. In other words, the median salary would be the salary where there are as many people earning more money as there are earning less money. If there are an even number of data points, the median is halfway between the two middle data points.
    1. Turn off the "Show mean" option, and turn on the "Show median" option. A vertical blue line will appear showing the value of the median.
    2. What happens when you drag a coin to a different column on the same side of the median?
    3. What happens when you drag a coin to the other side of the median?
    4. If you add a very high or very low value to the data set, which measure do you think would be more affected: the mean or the median? (Hint: both the mean and the median can be thought of as "balance points", but they balance the data differently.)
  1. The "mode" of a data set is the value or values which occur most often. In this scenario, the mode would be the salary that the greatest number of employees have.
    1. Turn off the "Show median" option, and turn on the "Show mode" option. One or more vertical green lines will appear, showing the mode(s) for this data set.
    2. Click the "New" button several times to generate different data sets. Notice that sometimes, the mode is a single value, and sometimes, there are several modes. When does a set of data have several modes?
    3. What happens when you drag a coin out of one of the 'mode' columns?
    4. By dragging coins, can you make a data set that has five modes?
  1. The "range" of a data set is the difference between the highest value and the lowest value. In this scenario, the range is how much more the highest paid employee earns than the lowest paid employee.
    1. Turn off the "Show mode" option, and turn on the "Show range" option. The range will be shown by a horizontal bar under the graph.
    2. Can you drag coins to different columns to make the data set's range smaller? Larger? Can you drag a coin without affecting the range?
    3. Can you create a data set with the same range but with a different highest salary?
  1. Click the "Clear" button, and try building your own data sets by dragging coins onto the graph.
    1. Can you find an example of a data set whose mean is greater than its median? Whose mean is less than its median?
    2. Give an example of a data set whose mean, median, and mode are all the same value.
    3. Does the median always have to be equal to at least one of the values in the data set? What about the mean? Can you create a data set whose mean and median are not equal to any of the markers on the graph?
    4. How would you create a data set with more than one mode?
    5. What does a set with no mode look like?
    6. How would you create a data set with a range of 20?
    7. Can you create a data set with a range of 20 and a median of 80?

Assessment Questions

  1. What is the missing value in the data set {20, 30, ?, 50} if the mean is 50?
    1. – 20
    2. 40
    3. 50
    4. 100
  2. What is the median of the data set {30, 10, 20, 40}?
    1. 10
    2. 15
    3. 20
    4. 25
  3. Which of the following data sets has a range of 40?
    1. {40, 10, 20, 40}
    2. {40, 10, 20, 50}
    3. {40, 10, 20, 80}
    4. {40, 10, 20, 90}
  4. Which one of the following data sets has more than one mode?
    1. {30, 50, 10, 20, 30, 40, 50}
    2. {40, 10, 70, 20, 30, 50, 70}
    3. {30, 40, 30, 20, 20, 40, 30}
    4. {10, 20, 30, 40, 50, 60, 70}

Algebra 1 iText

Page

Chapter 2
Solving Equations