Explanation

The harmonic mean helps reduce the impact of large outliers but not small outliers because:

• Median: Resistant to both large and small outliers since it depends only on the middle value(s)
• Arithmetic mean: Sensitive to both large and small outliers
• Harmonic mean: Calculated as the reciprocal of the arithmetic mean of reciprocals: $H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$

Large outliers have small reciprocals, which reduces their impact on the harmonic mean. However, small outliers have large reciprocals, which increases their impact.

Example: For dataset [1, 2, 3, 100]:

• Arithmetic mean = 26.5 (heavily influenced by 100)
• Harmonic mean = 2.9 (less influenced by 100)
• Median = 2.5 (unaffected by 100)

For dataset [0.1, 2, 3, 4]:

• Arithmetic mean = 2.275
• Harmonic mean = 0.56 (heavily influenced by 0.1)
• Median = 2.5 (unaffected by 0.1)