How to Calculate the Geometric Mean

Geometric mean is a mathematical concept that is related to, but easily confused with, the more commonly used arithmetic mean. To calculate the geometric mean, use one of the methods below.

Steps:

Two Numbers: Simple Method

1. Find the numbers you wish to average.
   
   
   • Ex. 2 and 32.
2. Multiply them together.
   
   
   • Ex. 2 x 32 = 64.
3. Calculate the square root of said number.
   
   
   • Ex. √64 = 8.

Two Numbers: Detailed Method

1. Plug your numbers into the equation below.
   
   If your numbers are 10 and 15, for example, plug in 10 for “first #” and 15 for “second #.”
   
   
   
2. Solve for X.
   Start by cross-multiplying, which means multiplying the pairs of numbers diagonal to one another and then setting the results on opposite sides of an = sign. Since X*X is X^2, your equation should look like: X^2 = (product of your other numbers).
   
   To solve for X, find the square root of your product. If you’re lucky, the results will be a whole number. If not, you can provide a decimal answer or leave your answer in square root form, depending on what your instructor prefers. The example below is in simplified square root form.
   
   
   

Three or More Numbers: Simple Method

1. Plug your numbers into the equation below.
   Mean = (a₁ × a₂ . . . a_n)^1/n
   • a₁ is your first number, a₂ is your second number, and so forth
   • n is the number of entries
2. Multiply the numbers (a₁, a₂, etc.) together.
3. Calculate the nth root of this number. This is the geometric mean.

Three or More Numbers: Detailed Method

1. Find the log of each number and add the logarithmic values together.
   
   Find the LOG button on your calculator. When you’re ready, type: (first number) LOG + (second number) LOG + (third number) LOG [+ log of additional numbers as necessary] =. Do not neglect to type = or the number you see will be the log of the most recent number, not the total.
   
   
   • Ex. log 7 + log 9 + log 12 = 2.878521796…
2. Divide the sum of the logarithmic values by the number of values you added. If you added the logs of three numbers, divide by three.
   
   
   • Ex. 2.878521796 / 3 = .959507265…
3. Find the antilog of your result. On your calculator, press the 2nd function (usually yellow) and then LOG to activate the secondary function of the log button, or the antilog. This resulting value is the geometric mean.
   
   
   • Ex. antilog .959507265 = 9.109766916. Therefore, the geometric mean of 7, 9, and 12 is 9.12.

Video:

Tips:

• Difference between arithmetic and geometric mean:
  • If you wanted the arithmetic mean of 3, 4 and 18, for example, you would add 3 + 4 + 18, then divide by 3 because there are three numbers. The result would be 25/3 or about 8.333..., which shows that if you had three values of 8.3333..., it would give the same total as the individual values of 3, 4, and 18. The arithmetic mean answers the question, "If all the quantities had the same value, what would that value have to be in order to add up to the same total?"
  • By contrast, the geometric mean answers the question, "If all the quantities had the same value, what would that value have to be in order to have the same product when multiplied?" So to find the geometric mean of 3, 4 and 18, we would multiply 3 x 4 x 18. This would give us 216. We would then take the cubic root (cubic root because there were three original numbers). The answer would be 6. In other words, since 6 x 6 x 6 = 3 x 4 x 18, 6 is the geometric mean of 3, 4 and 18.
• The geometric mean only applies to non-negative numbers. In word problems where using a geometric mean is appropriate, the scenario will usually not make sense with negative numbers.
• The geometric mean of any set of numbers is always less than or equal to the arithmetic mean of that set.