Question

Derive the mathematical properties of mean

Answer 1

To solve different types of problems on average we need to follow the properties of arithmetic mean.

Here we will learn about all the properties and proof the arithmetic mean showing the step-by-step explanation.

What are the properties of arithmetic mean?

The properties are explained below with suitable illustration.

Property 1:

If x is the arithmetic mean of n observations x1, x2, x3, . . xn; then  

(x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x) = 0.  

 

Now we will proof the Property 1:

We know that  

x = (x1 + x2 + x3 + . . . + xn)/n  

⇒ (x1 + x2 + x3 + . . . + xn) = nx. ………………….. (A)  

Therefore, (x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x)

= (x1 + x2 + x3 + . . . + xn) - nx

= (nx - nx), [using (A)].

= 0.  

Hence, (x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x) = 0.