## Mean and Median

### Instructions

Overview: Investigate properties of the mean and median of a distribution. Then test your knowledge.

Use the menu in the upper-left corner of the applet to switch between the investigate and identify windows.

Investigate window - Use this window to investigate the value of the mean and median for a distribution of numbers you create in the grid.

• Click on boxes in the grid to create a distribution of numbers.
• The number of filled cells in any column indicates the number of times the corresponding column value occurs in the distribution.
• Clicking on the topmost filled box in any column clears the column.
• Clicking on any unfilled box in the column fills the column to the selected cell.
• Use the spinners at the top to change the number of rows or columns.
• The mean of the distribution is indicated by the blue triangle.
• The median of the distribution is indicated by the red triangle.
• The mean and median indicators can be toggled on or off using the check boxes at the top.

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Choose "Show values" from the menu to display the numeric values of the mean and median.

Identify window - By default, the window displays a distribution of hypothetical test scores (out of 10 points). An orange triangle marks a point in the distribution. Use the buttons at the bottom to identify the position indicated by the orange marker as the mean, the median, both of these, or neither of these. Click the ?show solution? button to see the actual positions of the mean and median in the distribution.

Use the test scores button to generate a new hypothetical distribution of test scores and the income button to generate a distribution of hypothetical incomes (in thousands of dollars).

Activity 1

Use the investigate window to explore how changing the distribution affects the mean and median. Set the number of rows to 12 and the number of columns to 10, then respond to the following prompts:
1. The mean and the median are the same when all columns have the same number of entries. Can you create a distribution in which the number of entries in the columns varies, but the mean and the median are the same?
• What does this distribution look like?
• Can you create more than one?
2. What must be true about the distribution for the mean and the median to be the same?
3. Create a distribution in which the difference between mean and median is as large as you can make it.
• What does this distribution look like?
• Can you create more than one distribution that will produce this maximum difference between mean and median?
4. Under what conditions are the median and the mean the most different?

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Activity 2

Test your knowledge of the relationship between mean and median using the identify window. As you do this consider the following:

1. What is your strategy for determining whether the marker is at the mean, the median, both, or neither?
2. Under what conditions is it most difficult to discern between the mean and the median?
3. Are you able to correctly determine when the marker is at neither the mean nor the median?
• How do you do this?
• Under what conditions is this most difficult to do?
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