Lesson Difference between Arithmetic mean and Geometric mean

The difference between Arithmetic mean and Geometric mean

This lesson demonstrates the difference between Average (or Arithmetic mean) and Geometric mean that were introduced in two previous lessons.

If we have two numbers and , then Arithmetic mean is equal to

.

If and are positive, then Geometric mean of these numbers is equal to

.

You can see that the definitions are different.
Now you will see that the calculated values for the both means might be different.

Let's consider =2, =8.
Then Arithmetic mean of numbers 2 and 8 is .
Geometric mean of these numbers is .
You see the difference.

Let's consider another example: =4, =5.
Then Arithmetic mean of numbers 4 and 5 is .
Geometric mean of these numbers is (approximately).
Again, the difference is obvious.

Let's consider third example: =5, =5.
Then Arithmetic mean of numbers 5 and 5 is .
Geometric mean of these numbers is .
In this case Arithmetic mean is equal to Geometric mean.

Arithmetic mean of the two positive numbers is always greater than their Geometric mean, except the case when both numbers are equal. In this case Arithmetic mean is equal to Geometric mean.

Arithmetic mean (average), Geometric mean, and Median are the useful tools for statistical analysis.
Three tools, if used properly, allow quickly analyze data-sets.

Arithmetic mean characterizes the average value in data-set.

Geometric mean allows to estimate the average growth rate for processes with variable in time growth rate. It is used often in financial analysis, when analysts look at processes with compound interest. Examples of this are given in the lesson on Geometric Mean.

Median is used to quickly subdivide the whole current data-set in two halves and to attribute each given sample to one of two sub-sets.