Question

How do you find the Geometric mean and the Harmonic mean?

Answer 1

Hint: The mean of some values is the average of those values. The geometric mean of $n$ values is a mean which is found by taking the ${{n}^{th}}$ root of the product of those $n$ values. The harmonic mean of some values is a mean which is the reciprocal of the arithmetic mean of the reciprocals of those values.

Complete step by step solution:
Suppose that we are given with $n$ observations. The mean of these $n$ observations is the average of these observations.
We know that the arithmetic mean of a certain number of observations is the sum of the observations divided by the number of observations.
Suppose that the $n$ observations we have mentioned here are ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}.$
If we are asked to find the arithmetic mean of these observations, we will use the following formula,
$\Rightarrow \text{Arithmetic} \text{Mean}=\dfrac{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}{n}.$
Like arithmetic mean, there are geometric mean and harmonic mean in Mathematics and Statistics.
The geometric mean of $n$ observations is defined as the ${{n}^{th}}$ root of the product of the $n$ observations.
Let us suppose that we are given with the $n$ observations ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}.$
To find the geometric mean of these observations we have to use the formula given below:
$\Rightarrow \text{Geometric} \text{Mean}=\sqrt[n]{{{x}_{1}}.{{x}_{2}}...{{x}_{n}}}.$
The harmonic mean of $n$ observations is defined as the value obtained by dividing $n$ by the sum of reciprocals of the $n$ observations.
Let us define the harmonic mean as follows:
$\Rightarrow \text{Harmonic} \text{Mean}=\dfrac{n}{\dfrac{1}{{{x}_{1}}}+\dfrac{1}{{{x}_{2}}}+...+\dfrac{1}{{{x}_{n}}}}.$
Using the above formulas, like arithmetic mean, we can also find the geometric mean and the harmonic mean.

Note: The three means, the arithmetic mean, the geometric mean and the harmonic mean, are together called the Pythagorean means. From the formula, we can see that if any of the observations is $0,$ then the geometric mean is $0.$ Also, if any of the observations or odd number of observations is/are negative, then the geometric mean becomes imaginary. It is difficult to calculate the harmonic mean.