Harmonic Mean Calculator – Guide & Formulas
Calculate the harmonic mean of two or more numbers. Find the harmonic average for rates, ratios, and reciprocal-based data with step-by-step explanation.
Calculate the harmonic mean of any set of positive numbers. Enter values to instantly find the harmonic average, see the formula applied, and compare with arithmetic and geometric means.
Key Takeaway
Use the free Harmonic Mean Calculator to calculate the harmonic mean of two or more numbers. find the harmonic average for rates, ratios, and reciprocal-based data with step-by-step explanation. Get instant results with step-by-step explanations.
How to Use the Harmonic Mean Calculator
- Enter your set of positive numbers (comma-separated or one at a time).
- Review the computed harmonic mean.
- See the arithmetic mean and geometric mean for comparison.
- Understand when harmonic mean is preferred (rates, speeds, ratios).
- Use for problems involving rates, efficiency, or averages of reciprocals.
The Formula
Variable Definitions
- n: The number of values in the dataset
- x₁, x₂, ..., xₙ: The positive values in the dataset
- 1/xᵢ: The reciprocal of each value
- HM: Harmonic mean — the reciprocal of the arithmetic mean of reciprocals
Harmonic Mean of 4 and 8
Calculate the harmonic mean of 4 and 8, and compare with arithmetic and geometric means.
- Step 1: Identify the values: 4 and 8 (n = 2).
- Step 2: Compute reciprocals: 1/4 = 0.25 and 1/8 = 0.125.
- Step 3: Sum the reciprocals: 0.25 + 0.125 = 0.375.
- Step 4: Divide n by the sum: HM = 2 / 0.375 = 5.333.
- Step 5: Compare: HM = 5.333, GM = √(4×8) = 5.657, AM = (4+8)/2 = 6.
- Step 6: The harmonic mean (5.333) is the smallest of the three means, as expected for unequal numbers.
Frequently Asked Questions
What is the harmonic mean?
The harmonic mean of n positive numbers is the reciprocal of the arithmetic mean of their reciprocals: HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ). It is the smallest of the three Pythagorean means (HM ≤ GM ≤ AM) for unequal positive numbers. The harmonic mean is especially useful for averaging rates.
When should I use the harmonic mean?
Use harmonic mean when: (1) Averaging rates or ratios (speed, efficiency, density), (2) Each value represents a rate with different units or denominators, (3) You need to average prices paid per unit at different quantities, (4) Comparing parallel resistances in circuits, (5) Computing average speed for equal-distance trips.
What is the difference between harmonic mean and arithmetic mean?
Arithmetic mean = sum/n. Harmonic mean = n/sum(reciprocals). AM is better for additive data, HM for rate data. For example: driving 60 mph for 100 miles then 120 mph for 100 miles: AM = 90 mph, but HM = 2/(1/60 + 1/120) = 80 mph. HM correctly accounts for the time spent at each speed.
How do I calculate the weighted harmonic mean?
Weighted harmonic mean = Σwᵢ / Σ(wᵢ/xᵢ), where wᵢ are weights. For example, if you drive 100 miles at 50 mph and 200 miles at 100 mph: WHM = (100+200) / (100/50 + 200/100) = 300 / (2+2) = 75 mph.
What is the relationship between harmonic mean and parallel resistors?
The equivalent resistance of resistors in parallel equals the harmonic mean divided by the number of resistors. For two resistors R₁ and R₂ in parallel: R_eq = (R₁ × R₂) / (R₁ + R₂) = HM(R₁, R₂) / 2.
Why is harmonic mean used for average speed?
When traveling equal distances at different speeds, the average speed is the harmonic mean of the speeds. This is because time = distance/speed, and you are averaging the times per unit distance (the reciprocals of speed). For equal distances: HM = 2ab/(a+b).
Can the harmonic mean handle zero or negative numbers?
No. The harmonic mean requires all positive numbers because it involves reciprocals. Division by zero is undefined, and negative reciprocals would produce incorrect averages. If your data contains zeros or negatives, use the arithmetic mean instead.
What is the formula for harmonic mean of two numbers?
For two numbers a and b: HM = 2ab / (a + b). This is also the formula for the equivalent resistance of two parallel resistors. It is equivalent to the reciprocal of the average of 1/a and 1/b.
How does harmonic mean relate to the other Pythagorean means?
For any set of positive numbers (not all equal): HM < GM < AM. This is the Pythagorean means inequality. Equality holds only when all values are the same. The geometric mean is the geometric mean of the harmonic and arithmetic means: GM = √(HM × AM) for two numbers.