# Properties of Arithmetic Mean

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 x

_{1}, x

_{2}, x

_{3}, . . x

_{n}; then

(x

_{1} -

x) + (x

_{2} -

x) + (x

_{3} -

x) + ... + (x

_{n} -

x) = 0.

**Now we will proof the Property 1:**

We know that

x = (x

_{1} + x

_{2} + x

_{3} + . . . + x

_{n})/n

⇒ (x

_{1} + x

_{2} + x

_{3} + . . . + x

_{n}) = n

x. ………………….. (A)

Therefore, (x

_{1} -

x) + (x

_{2} -

x) + (x

_{3} -

x) + ... + (x

_{n} -

x)

= (x

_{1} + x

_{2} + x

_{3} + . . . + x

_{n}) - n

x
= (n

x - n

x), [using (A)].

= 0.

Hence, (x

_{1} -

x) + (x

_{2} -

x) + (x

_{3} -

x) + ... + (x

_{n} -

x) = 0.

Property 2:

The mean of n observations x

_{1}, x

_{2}, . . ., x

_{n} is

x. If each observation is increased by p, the mean of the new observations is (

x + p).

**Now we will proof the Property 2:**

x = (x

_{1} + x

_{2} +. . . + x

_{n})/n

⇒ x

_{1} + x

_{2} + . . . + x

_{n}) = n

x …………. (A)

Mean of (x

_{1} + p), (x

_{2} + p), ..., (x

_{n} + p)

= {(x

_{1} + p) + (x

_{2} + p) + ... + (x

_{1} + p)}/n

= {(x

_{1} + x

_{2} + …… + x

_{n}) + np}/n

= (n

x + np)/n, [using (A)].

= {n(

x + p)}/n

= (

x + p).

Hence, the mean of the new observations is (

x + p).

Property 3:

The mean of n observations x

_{1}, x

_{2}, . . ., x

_{n} is

x. If each observation is decreased by p, the mean of the new observations is (

x - p).

**Now we will proof the Property 3:**

x = (x

_{1} + x

_{2} +. . . + x

_{n})/n

⇒ x

_{1} + x

_{2} + . . . + x

_{n}) = n

x …………. (A)

Mean of (x

_{1} - p), (x

_{2} - p), ...., (x

_{n} - p)

= {(x

_{1} - p) + (x

_{2} - p) + ... + (x

_{1} - p)}/n

= {(x

_{1} + x

_{2} + …. + x

_{n}) - np}/n

= (n

x - np)/n, [using (A)].

= {n(

x - p)}/n

= (

x - p).

Hence, the mean of the new observations is (

x + p).

Property 4:

The mean of n observations x

_{1}, x

_{2}, . . .,x

_{n} is

x. If each observation is multiplied by a nonzero number p, the mean of the new observations is p

x.

**Now we will proof the Property 4: **

x = (x

_{1} + x

_{2} + . . . + x

_{n})/n

⇒ x

_{1} + x

_{2} + . . . + x

_{n} = n

x …………… (A)

Mean of px

_{1}, px

_{2}, . . ., px

_{n},

= (px

_{1} + px

_{2} + ... + px

_{n})/n

= {p(x

_{1} + x

_{2} + ... + x

_{n})}/n

= {p(n

x)}/n, [using (A)].

= p

x.

Hence, the mean of the new observations is p

x.

Property 5:

The mean of n observations x

_{1}, x

_{2}, . . ., x

_{n} is

x. If each observation is divided by a nonzero number p, the mean of the new observations is (

x/p).

**Now we will proof the
Property 5:**

x = (x

_{1} + x

_{2} + ... + x

_{n})/n

⇒ x

_{1} + x

_{2} + ... + x

_{n}) = n

x …………… (A)

Mean of (x

_{1}/p), (x

_{2}/p), . . ., (x

_{n}/p)

= (1/n) ∙ (x

_{1}/p + x

_{2}/p + …. x

_{n}/p)

= (x

_{1} + x

_{2} + ... + x

_{n})/np

= (n

x)/(np), [using (A)].

= (

x/p).

To get more ideas students can follow the below links to
understand how to solve various types of problems using the properties of
arithmetic mean.

**9 Grade Math**

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