A data set consists of the values 10, 12 and 15. What is the harmonic mean of these values, to three decimals?
Correct Answer
B) 12.000
H=3/(1/10+1/12+1/15)=12. The arithmetic and geometric means use different definitions.
Why This Is the Correct Answer
Option B is correct because H = 3 / (1/10 + 1/12 + 1/15) = 3 / 0.25 = 12.000. The reciprocals sum exactly to 0.25 (6/60 + 5/60 + 4/60 = 15/60), so the harmonic mean is exactly 12. This follows the definition of the harmonic mean, HM = n / Σ(1/Xi).
Why the Other Options Are Wrong
Option A: 12.333
Option A, 12.333, is the arithmetic mean: (10 + 12 + 15) / 3 = 37 / 3 = 12.333. The arithmetic mean uses a different definition; the harmonic mean is 3 / 0.25 = 12.000.
Option C: 12.164
Option C, 12.164, is the geometric mean: the cube root of 10 × 12 × 15 = 1,800, which is about 12.164. That is a different definition; the harmonic mean formula gives 3 / 0.25 = 12.000.
Option D: 12.500
Option D, 12.500, is the mid-range, the average of the smallest and largest values: (10 + 15) / 2 = 12.500. It ignores the middle value and the reciprocal step; the harmonic mean is 3 / (1/10 + 1/12 + 1/15) = 3 / 0.25 = 12.000.
Flip, add, divide
Harmonic mean: flip each value to its reciprocal, add the reciprocals, then divide the count by that total. For unequal positive values the order is harmonic below geometric below arithmetic.
How to use: Use this ordering to spot which distractor is the arithmetic or geometric mean.
Exam Tip
For unequal positive values, the harmonic mean comes out below both the geometric and arithmetic means, so several nearby averages will appear among the choices; compute n / Σ(1/x) rather than picking the closest-looking figure.
Common Mistakes to Avoid
- -Computing the arithmetic mean (12.333) instead of the harmonic mean
- -Computing the geometric mean (12.164) by mistake
- -Averaging only the smallest and largest values (the mid-range)
Concept Deep Dive
Analysis
This question tests whether you can apply the harmonic mean formula H = n / Σ(1/x) and keep it distinct from the arithmetic and geometric means. Take the reciprocal of each value, add them, and divide the count by that sum. The reciprocals 0.1, 0.08333 and 0.06667 total 0.25, and 3 / 0.25 = 12.000.
Background Knowledge
The harmonic mean is the count of values divided by the sum of their reciprocals. For positive values that are not all equal, the harmonic mean is lower than the geometric mean, which is lower than the arithmetic mean. Which mean is appropriate depends on the problem, so none is preferred for every appraisal use.
Real-World Application
An analyst averaging price-per-unit ratios across several sales notices that ratios expressed as reciprocals call for a harmonic mean. The analyst computes it with n divided by the sum of reciprocals and reports which mean was used. Comparing it with the arithmetic mean shows how much the choice of average affects the figure.
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