**Properties of arithmetic mean:**

Here, we are going to see the properties of arithmetic mean.

Arithmetic mean is one of the measures of central tendency which can be defined as the sum of all observations to be divided by the number of observations.

Now, let us look at the properties of arithmetic mean.

**Property 1 : **

If all the observations assumed by a variable are constants, say "k", then arithmetic mean is also "k".

For example, if the height of every student in a group of 10 students is 170 cm, the mean height is, of course 170 cm.

**Property 2 :**

The algebraic sum of deviations of a set of observations from their arithmetic mean is zero.

That is,

for unclassified data, ∑(x - x̄) = 0.

And for a grouped frequency distribution, ∑f(x - x̄) = 0.

For example, if a variable "x" assumes five observations, say 10, 20, 30, 40, 50, then x̄ = 30.

The deviations of the observations from arithmetic mean (x - x̄) are -20, -10, 0, 10, 20.

Now, ∑(x - x̄) = (-20) + (-10) + 0 + 10 + 20 = 0

**Property 3 :**

Arithmetic mean is affected due to a change of origin and/or scale which implies that if the original variable "x" is changed to another variable "y" effecting a change of origin, say "a" and scale, say "b", of "x". That is y = a + bx.

Then we have,

Arithmetic mean of "y" = a + bx̄

For example, if it is known that two variables x and y are related by 2x + 3y + 7 = 0 and x̄ = 15, then

Arithmetic mean of "y" = (-7 - 2x̄) / 3

Plug x̄ = 15

Arithmetic mean of "y" = (-7 - 2x15) / 3

Arithmetic mean of "y" = (-7 - 30) / 3

Arithmetic mean of "y" = -37/ 3

Arithmetic mean of "y" = -12.33

**Property 4 : **

If there are two groups containing n₁ and n₂ observations

x̄₁ and x̄₂ are the respective arithmetic means, then the combined arithmetic mean is given by

**x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)**

This property could be extended to more than two groups and we may write it as

**x̄ = ∑nx̄ / ∑n**

Here,

∑nx̄ = n₁x̄₁ + n₂x̄₂ + ..............

∑n = n₁ + n₂ + ........................

1) It is rigidly defined.

2) It is based on all the observations.

3) It is easy to comprehend.

4) It is simple to calculate.

5) It is least affected by the presence of extreme observations.

6) It is amenable to mathematical treatment or properties.

The above properties make "Arithmetic mean" as the best measure of central tendency.

However, arithmetic mean has some draw backs.

They are,

1) It is very much affected by sampling fluctuation.

2) Arithmetic mean can not be advocated to open en classification.

For open end classification, the most appropriate measure of central tendency is "Median.

After having gone through the stuff given above, we hope that the students would have understood "Properties of arithmetic-mean".

Apart from the stuff given above, if you want to know more about "Properties of arithmetic-mean",please click here

Apart from the stuff given on this web page, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**