Question

how to find harmonic mean

Answer 1

The harmonic mean is found by dividing the number of observations by the sum of the reciprocals of the observations.

Answer 2

To find the harmonic mean of a set of numbers, you take the total count of the numbers and divide it by the sum of the reciprocals of each number.

For a set of $n$ observations ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}$, the Harmonic Mean (HM) is given by the formula:

$$\text{Harmonic Mean} = \frac{n}{\frac{1}{{{x}_{1}}} + \frac{1}{{{x}_{2}}} + \dots + \frac{1}{{{x}_{n}}}}$$

Example:

To find the harmonic mean of 2, 3, and 5: Here, $n=3$, ${{x}_{1}}=2$, ${{x}_{2}}=3$, ${{x}_{3}}=5$.

$$\text{HM} = \frac{3}{\frac{1}{2} + \frac{1}{3} + \frac{1}{5}}$$ $$\text{HM} = \frac{3}{\frac{15+10+6}{30}}$$ $$\text{HM} = \frac{3}{\frac{31}{30}}$$ $$\text{HM} = \frac{3 \times 30}{31}$$ $$\text{HM} = \frac{90}{31} \approx 2.903$$

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Explanation of the solution:

The harmonic mean of $n$ observations ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}$ is calculated as the reciprocal of the arithmetic mean of the reciprocals of the observations. The formula is:

$$\text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$