The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals. Which statement is true?

The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals. Which statement is true?

• A. It equals the arithmetic mean when all values are equal.
• B. It equals the maximum value in the set.
• C. It equals the minimum value in the set.
• D. It is the sum of all values.

Answer: (A)

The key idea here is how the harmonic mean relates to the arithmetic mean, specifically when they coincide. If every value in the data set is the same, say each equals c, then the reciprocals are all 1/c and their average is 1/c. Taking the reciprocal of that average gives c, which is exactly the arithmetic mean in this case as well. So the harmonic mean equals the arithmetic mean precisely when all values are equal. In general, the harmonic mean is less than or equal to the arithmetic mean, with equality only in that equal-values situation. The other statements aren’t universally true: the harmonic mean isn’t generally the maximum or the minimum, nor is it the sum of the values.

The key idea here is how the harmonic mean relates to the arithmetic mean, specifically when they coincide. If every value in the data set is the same, say each equals c, then the reciprocals are all 1/c and their average is 1/c. Taking the reciprocal of that average gives c, which is exactly the arithmetic mean in this case as well. So the harmonic mean equals the arithmetic mean precisely when all values are equal. In general, the harmonic mean is less than or equal to the arithmetic mean, with equality only in that equal-values situation. The other statements aren’t universally true: the harmonic mean isn’t generally the maximum or the minimum, nor is it the sum of the values.