Harmonic Mean
Reciprocal of the average of reciprocals.
Formula
Variables
Example
HM of 1,2,4 = 1.71
How it works
The harmonic mean is an average designed for quantities that combine through reciprocals, so each value contributes according to 1/x_i rather than directly. This makes it especially sensitive to small values: a single low number pulls the result down strongly, which matches situations where a bottleneck or slow rate limits the whole process. It comes from asking for the reciprocal of the arithmetic mean of the reciprocals, so it naturally describes averages of rates and ratios.
When to use it
Use the harmonic mean when averaging rates over the same fixed quantity, such as speeds over equal distances or cost-per-unit values over equal amounts. It is also appropriate in physics, engineering, and finance whenever the correct “average” must reflect reciprocal relationships rather than ordinary addition.
Worked example, step by step
Find the harmonic mean of 3, 6, and 12. Step 1: Count the values: n = 3. Step 2: Take reciprocals: 1/3, 1/6, 1/12. Step 3: Add the reciprocals: 1/3 + 1/6 + 1/12 = 4/12 + 2/12 + 1/12 = 7/12. Step 4: Apply the formula: HM = n / (sum of reciprocals) = 3 / (7/12). Step 5: Divide by a fraction: 3 × 12/7 = 36/7. Step 6: Convert to decimal: HM ≈ 5.14. Final answer: HM = 36/7 ≈ 5.14.
Common mistakes to avoid
- Using the arithmetic mean instead of averaging the reciprocals first
- Forgetting that the formula depends on the number of values n
- Applying the harmonic mean to quantities that are not rates or reciprocal-type measurements
Did You Know?
The harmonic mean gives the correct average speed for a round trip at two different speeds.
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