Question

How are various types of mean like Arithmetic mean, Geometric mean, and Harmonic mean different?

How are they calculated?

Answer 1

Arithmetic mean of a set of n observations is the sum of the values of the observations, divided by the number of observations. 

If \( x_1, x_2, x_3,.......x_n \) are n observations then:

Arithmetic Mean (A.M) = \( {\frac {x_{1}+x_{2}+x_{3}+\cdots +x_{n}}{n}}} \)

Arithmetic Mean (A.M) = \( {\frac {1}{n}}\sum _{i=1}^{n}x_{i} \)

Arithmetic Mean is highly affected by the presence of outliers or when observations follow non-Gaussian distribution.

Geometric mean of a set of n observations is the n-th root of their product. 

If \( x_1, x_2, x_3,.......x_n \) are n observations then:

Geometric Mean (G.M) = \( {\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}} \)

Geometric Mean (G.M) = \( \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}} \)

Geometric mean is used when the data contains different units or when relative changes rather than absolute values is our interest.

Harmonic mean of a set of n observations is the  reciprocal of the arithmetic mean of their reciprocals.

If \( x_1, x_2, x_3,.......x_n \) are n observations then:

Harmonic Mean (H.M) = \( {\frac {1}{\begin{array}{l}{\frac{1}{n}[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+……………+\frac{1}{x_{n}}]}\end{array}}}} \)

Harmonic Mean (H.M) =\( {\frac {1}{\begin{array}{l}{\frac{1}{n}}[{\sum_{i=1}^{n}\frac{1}{x_{i}}]}\end{array}}}} \)

Harmonic mean is used in situations like when the data is in the form of rates.

Arithmetic mean of a set of n observations is the sum of the values of the observations, divided by the number of observations. Geometric mean of a set of n observations is the n-th root of their product. Harmonic mean of a set of n observations is the  reciprocal of the arithmetic mean of their reciprocals.