# 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 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.

Property 2:

The mean of n observations x1, x2, . . ., xn 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 = (x1 + x2 +. . . + xn)/n

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

Mean of (x1 + p), (x2 + p), ..., (xn + p)

= {(x1 + p) + (x2 + p) + ... + (x1 + p)}/n

= {(x1 + x2 + …… + xn) + np}/n

= (nx + 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 x1, x2, . . ., xn 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 = (x1 + x2 +. . . + xn)/n

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

Mean of (x1 - p), (x2 - p), ...., (xn - p)

= {(x1 - p) + (x2 - p) + ... + (x1 - p)}/n

= {(x1 + x2 + …. + xn) - np}/n

= (nx - 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 x1, x2, . . .,xn is x. If each observation is multiplied by a nonzero number p, the mean of the new observations is px.

Now we will proof the Property 4:

x = (x1 + x2 + . . . + xn)/n

⇒ x1 + x2 + . . . + xn = nx …………… (A)

Mean of px1, px2, . . ., pxn,

= (px1 + px2 + ... + pxn)/n

= {p(x1 + x2 + ... + xn)}/n

= {p(nx)}/n, [using (A)].

= px.

Hence, the mean of the new observations is px.

Property 5:

The mean of n observations x1, x2, . . ., xn 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 = (x1 + x2 + ... + xn)/n

⇒ x1 + x2 + ... + xn) = nx …………… (A)

Mean of (x1/p), (x2/p), . . ., (xn/p)

= (1/n) ∙ (x1/p + x2/p + …. xn/p)

= (x1 + x2 + ... + xn)/np

= (nx)/(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.

Statistics

Word Problems on Arithmetic Mean

Properties of Arithmetic Mean

Problems Based on Average

Properties Questions on Arithmetic Mean