Question

Harmonic mean gives more weightage to :

• A. Small values
• B. Large values
• C. Positive values
• D. Negative values

Hint:

Write HM as \(n / \sum(1/x_i)\) and check which values make \(1/x_i\) large.

Answer 1

The Correct Option is A

Step 1: Recall the formula for harmonic mean of \(n\) observations \(x_1, x_2, \ldots, x_n\).\[HM = \frac{n}{\sum \frac{1}{x_i}}\]

Step 2: Look at how the reciprocals behave. If \(x_i\) is small, then \(\frac{1}{x_i}\) is large. If \(x_i\) is large, then \(\frac{1}{x_i}\) is small.

Step 3: The sum \(\sum \frac{1}{x_i}\) is therefore dominated by the reciprocals coming from the small values in the data, since these terms are the biggest contributors.

Step 4: Because the harmonic mean is built from this sum of reciprocals, the small values end up pulling the value of HM toward themselves. So HM stays close to the smaller observations and is always less than or equal to the geometric mean and arithmetic mean of the same data.

Step 5: This means the harmonic mean gives more weightage to small values. The correct option is (A) Small values.