Arithmetic/Geometric Means Essay

by Mike Callinan

This essay will explore the relationship between arithmetic and geometric means. First, we will prove that the arithmetic means is always greater than or equal to the geometric means by an algebraic proof. Then, we will visually display and demonstrate this relationship by spreadsheet using Microsoft Excel and two sketches using "Geometric Sketch Pad, Enhanced Version 3.00".

Algebraic Proof:

Given two values X and Y, both being greater than or equal to zero, we will prove that the arithmetic mean ( ) is greater than or equal to the geometric mean ( ). We know that the following is true:

Example Using Microsoft Excel Spreadsheet:

A comparison of arithmetic mean and geometric mean of a set a values for a and b can be easily visualized in a spreadsheet. The following is an example of a couple of rows from the attached Excel spreadsheet depicting values for a and b stored in columns A and B, respectively. The arithmetic mean is calculated in column C, with the geometric mean stored in column D. Example:

                 A                B                C                D                
      1.         1               10              5.5       3.16227766       
      2.         5               99               52       22.2485955       
      3.        20               20               20               20

Spreadsheet Attachment (Select to load the Excel Spreasheet with more data .).

Use of Geometry Sketchpad:

Circle:

Through use of Geometry Sketchpad, two segments of length A and B can be used to demonstrate that the arithmetic mean is always greater than the geometric mean except when A and B are equal (both means are equal under this condition) . The following sketch shows the two segments with and a circle created with a diameter of the combined segments. The radius CD represents the arithmetic mean . By dropping a perpendicular segment from the point D on the circle to point E on the line m, we get a right trianlge with side CE having a length of . By knowing the length of the hypotenuse CD and one of the sides CE, we can calculate the length of the other side DE as follows:

This turns out to be the geometric mean.

When segment B and segment A are equal in length, the following sketch shows that arithmetic and geometric mean are equal.

GSP Diameter Demonstration.If you have GSP Enhanced, Verison 3, this will load the above sketch with animation that will demonstrate the full range of values for segments A and B.

Rectangle Demonstration:

By taking those same two segments, A and B, we can create a sqare with sides of the combined lengths of A and B. The total area of the square will be while the volume of the four rectangles with sides A and B are . The former volume will always be greater than or at best, equal to the latter as shown algebraically as follows:

Once again, we have the left side of the equation representing the aritmetic mean and the right side representing the geometric mean. Please launch the following sketch (by selecting) to see how the square is set up.

GSP Rectangle Demonstration.

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