# PROPERTIES OF ARITHMETIC MEAN

## About "Properties of arithmetic mean"

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.

## 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 + b

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

and ₂ are the respective arithmetic means, then the combined arithmetic mean is given by

x̄  =  (n₁ + n₂) / (n₁ + n₂)

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

x̄  =  nx̄ / ∑n

Here,

nx̄  =  n₁ + n₂ + ..............

∑n  =  n₁ + n₂ + ........................ ## Some other properties of arithmetic mean

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

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6