Tutorial Three: Averages… Part 2!

Averages are extremely powerful mathematical tools.  In the previous tutorial, you were introduced to the addition mean equation, which is used to find the average (or arithmetic mean) of two numbers. Most people automatically use the addition mean equation when asked to find the average of two numbers. But, did you know that there is another novel way to find an average?

Imagine once again that you are asked to find the average of  7 and 3. You already know that the answer is 5. But this time, your equation cannot use the addition operator, also known as the plus “+” sign. This means that you can’t use the addition mean equation. So, let’s explore how we might accomplish this.

Conceptually, we know that if we were to plot the numbers along a line, the average would be the value that falls at the mid-point between both numbers.  Knowing this, we can find the average without using the plus sign. Begin by taking the difference of the numbers, which in this case is four because 7–3=4. Next, divide that difference by two, which produces two since 4÷2=2. Since this value is one–half of the difference, it is called the half-difference. We’re almost done.

Since the average is the value at the mid–point between both original numbers, you have a choice. First, you could add the half–difference to the smaller number to find the average, which is 3+2=5. But wait a minute; we said that you couldn’t use the plus sign. So, instead, you use the second option to find the average, which is to subtract the half–difference from the larger number.  In other words, you’ve just found the average without using the plus sign as: 7–2=5.

This approach, which uses subtraction to find an average of two numbers is called the subtraction mean method. Mathematically, it is written as:

Avg = A–(A–B)/2

It doesn’t matter if you let A=3 and B=7 or you let A=7 and B=3.  Both will produce 5 as the answer.

It is quite easy to show that the subtraction mean equation and the addition mean equation will always produce the same answers. In other words, the subtraction mean equation and the addition mean equation are equivalent equations that are simply written in different forms. While it is an extremely important equation, it is unfortunate that the subtraction mean equation isn’t always immediately recognized as being an average.

Fact: The average when expressed in the form of the subtraction mean equation is one of the most important foundational equations in physics.


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