Question

Calculate the Harmonic Mean of the following data:
10 \quad 20 \quad 40 \quad 60 \quad 120

• A. 25
• B. 27.5
• C. 32.5
• D. 50

Hint:

For any set of positive numbers, the Harmonic Mean \(\leq\) Geometric Mean \(\leq\) Arithmetic Mean. For this dataset, the Arithmetic Mean is (10+20+40+60+120)/5 = 50. The HM must be less than 50. This can help eliminate incorrect higher options.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The harmonic mean is a type of average, calculated as the reciprocal of the arithmetic mean of the reciprocals of the observations. It is particularly useful for averaging rates.

Step 2: Key Formula or Approach:
The formula for the Harmonic Mean (HM) is: \[ \text{HM} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \] where \( n \) is the number of observations and \( x_i \) are the individual values.

Step 3: Detailed Calculation:
1. Identify the data and n: The data points are 10, 20, 40, 60, 120. The number of observations, \(n\), is 5.
2. Calculate the reciprocals of each value:
\( \frac{1}{10}, \frac{1}{20}, \frac{1}{40}, \frac{1}{60}, \frac{1}{120} \)
3. Sum the reciprocals (\(\sum \frac{1}{x_i}\)):
To add these fractions, we find a common denominator, which is 120.
\[ \sum \frac{1}{x_i} = \frac{12}{120} + \frac{6}{120} + \frac{3}{120} + \frac{2}{120} + \frac{1}{120} = \frac{12+6+3+2+1}{120} = \frac{24}{120} \] Simplifying the fraction: \( \frac{24}{120} = \frac{1}{5} \) or 0.2.
4. Calculate the Harmonic Mean:
\[ \text{HM} = \frac{n}{\sum \frac{1}{x_i}} = \frac{5}{\frac{1}{5}} = 5 \times 5 = 25 \]
Step 4: Final Answer:
The Harmonic Mean of the given data is 25. Therefore, option (A) is correct.