Calculate Harmonic Mean from the following data:
10, 15, 20 and 25

Question

Calculate Harmonic Mean from the following data: 
10, 15, 20 and 25

Accepted Answer

To calculate the Harmonic Mean (HM) from the given data (10, 15, 20, 25), we use the formula:

HM = n / (Σ(1/x_i))

Where:
*   n is the number of observations.
*   x_i represents each individual observation.
*   Σ(1/x_i) is the sum of the reciprocals of the observations.

**1. Justification for Option 2 (15.58):**

Given data: 10, 15, 20, 25
Number of observations (n) = 4

First, calculate the reciprocal of each observation:
*   1/10 = 0.1
*   1/15 ≈ 0.06666667
*   1/20 = 0.05
*   1/25 = 0.04

Next, sum these reciprocals (Σ(1/x_i)):
Σ(1/x_i) = 0.1 + 0.06666667 + 0.05 + 0.04
Σ(1/x_i) = 0.25666667

To maintain precision, it's better to use fractions for the sum:
LCM of 10, 15, 20, 25 is 300.
1/10 = 30/300
1/15 = 20/300
1/20 = 15/300
1/25 = 12/300

Σ(1/x_i) = (30 + 20 + 15 + 12) / 300 = 77 / 300

Now, apply the Harmonic Mean formula:
HM = n / (Σ(1/x_i))
HM = 4 / (77/300)
HM = 4 * (300 / 77)
HM = 1200 / 77

Calculating the decimal value:
HM ≈ 15.58441558...

Rounding to two decimal places, the Harmonic Mean is 15.58.
Therefore, Option 2 is the correct answer.

**2. Explanation for why other options are incorrect:**

*   **Option 1 (17.5):**
    This value represents the Arithmetic Mean (AM) of the data.
    AM = (10 + 15 + 20 + 25) / 4 = 70 / 4 = 17.5.
    While 17.5 is the Arithmetic Mean, the question specifically asks for the Harmonic Mean. For a set of positive numbers that are not all equal, the Harmonic Mean is always less than the Arithmetic Mean (HM < AM). Since 15.58 < 17.5, this confirms that 17.5 is not the Harmonic Mean.

*   **Option 3 (1.75):**
    This value is significantly smaller than the actual Harmonic Mean. It appears to be 1/10th of the Arithmetic Mean (17.5 / 10 = 1.75) or a result of a major calculation error. It does not correspond to the correct calculation of the Harmonic Mean.

*   **Option 4 (1.55):**
    This value is also significantly smaller than the actual Harmonic Mean. It is approximately 1/10th of the correct Harmonic Mean (15.58 / 10 ≈ 1.558) or a result of a major calculation error. It does not correspond to the correct calculation of the Harmonic Mean.