types of mean in arithmetic

types of mean in arithmetic. Apart from arithmetic mean, there are other types of means occasionally used for calculation. These are geometric means, the harmonic mean and the quadratic mean.

TYPES OF MEAN IN ARITHMETIC

geometric means,

the harmonic mean and

the quadratic mean.

H

     GEOMETRIC MEAN

Meaning: The geometric mean of a group of numbers is the Nth root of the product of the numbers. In other words, it is derived from a set of N observations by taking the Nth root of the product of the numbers. It is denoted by letter G.

Formula for calculating geometric mean

The formula used for calculating geometric mean is:

G =        Nth    Product of the various

                        values in an observation

G =          N     x₁ + x₂ + x₃ ……….xn

Where x = individual value

Example

Calculate the geometric mean of the following set of data: 6, 8, 12

Solution

n = 3

Product of the various values

= 6 x 8 x 12 = 576

 G =         N     x₁ + x₂ + x₃

                3      6 + 8 + 12

                3      576

G = 8.32

Advantages of geometric mean

1. All available data are used in calculating geometric mean
2. It is useful in calculating statistical data.
3. It provides a balanced information on both sides of the distribution
4. It is very important in carrying out research work

Disadvantages of geometric mean

1. Involvement of large volume of data makes geometric mean difficult to understand
2. Geometric mean is at times difficult to compute.

HARMONIC MEAN

Meaning: Harmonic mean refers to the  reciprocal of the arithmetic mean of the reciprocal of some given numbers. It could have numbers like X₁, X₂, X₃, ………………..X₁₂. The harmonic mean is denoted by letter H.

Formula for calculating harmonic mean

The formula used for calculating harmonic mean is given below.

                                    N

Harmonic mean, H = Ʃ ¹/₈

Example

Calculate the harmonic mean of the following set of data 4, 6, and 8

Solution

The reciprocal of 4, 6 and 8 are ¼, and ¹/₈

The arithmetic mean of the reciprocal are

H = ¹/₃ ( ¼ + ¹/₆ + ¹/₈)

= ¹/₃ ( 6 + 4 + 3)

               24

= ¹/₃     13        =          13

            24                    72

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of given numbers

H =      72 = 54

            13

Advantages of harmonic mean

1. Harmonic mean can easily be determined
2. It does not affect the extremes of values in given data
3. All values in the observation are taken into consideration

Disadvantages of harmonic mean

1. Harmonic mean principles are difficult to understand
2. Its scope is limited
3. It is difficult to calculate

QUADRATIC MEAN

Meaning: Quadratic mean, also known as the root mean square (RMS), refers to the square root of the arithmetic mean if their squares. The quadratic mean is represented by R.M.S.

The formula used for calculating quadratic mean is:

R.M.S  =          Ʃx²

                        N

Example

Calculate the quadratic mean of the following set of numbers: 2, 4, 6, 8

Solution

R.M.S =          22 + 42 + 62 + 82

                                    4

            =          4 + 16 + 36 + 64

                                    4

            =          120

                          4                    =          30

                                                =          5.4

Advantages of quadratic mean

1. All values in a given data are taken in consideration
2. It is easy to determine

Disadvantages of quadratic mean

1. Calculation becomes very difficult when given values are large
2. Its principles are difficult to understand

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