Case 1
- First value (a)
- 2
- Second value (b)
- 3
2.400000
Open with these valuesHarmonic Mean of Two Numbers
2.400000
Result: 2.400000For two values the harmonic mean is 2ab ÷ (a + b): for 2 and 3 that is 12 ÷ 5 = 2.4. This calculator takes exactly two numbers, not a whole list. Drive equal distances at 40 and 60 km/h and your average speed is 48, not 50 — that gap is what the harmonic mean is for.
Worked examples
Case 2
- First value (a)
- 40
- Second value (b)
- 60
48.000000
Open with these valuesCase 3
- First value (a)
- 10
- Second value (b)
- 20
13.333333
Open with these valuesHow it's calculated
H = 2ab ÷ (a + b)
- StepEnter two positive numbers; zero and negatives have no harmonic mean.
- StepMultiply them and double the product.
- StepDivide that by the sum of the two values.
- ResultCompare with the arithmetic mean (a + b) ÷ 2 — it is never larger.
Reference table
| a, b | Arithmetic mean | Harmonic mean |
|---|---|---|
| 1, 1 | 1 | 1 |
| 0.5, 2 | 1.25 | 0.8 |
| 2, 3 | 2.5 | 2.4 |
| 3, 6 | 4.5 | 4 |
| 10, 20 | 15 | 13.333333 |
| 40, 60 | 50 | 48 |
Questions
How do I calculate the harmonic mean of two numbers?
Multiply the two values, double the result and divide by their sum: H = 2ab ÷ (a + b). The harmonic mean of 2 and 3 is 12 ÷ 5 = 2.4.
What is the harmonic mean?
It is the reciprocal of the average of the reciprocals, which for two numbers simplifies to 2ab ÷ (a + b). It is the correct way to average rates and ratios, most famously the average speed of a trip that covers equal distances at two speeds.
How does it differ from the arithmetic mean?
The arithmetic mean is the plain (a + b) ÷ 2. The harmonic mean gives more weight to the smaller value, so it is always less than or equal to the arithmetic one, equal only when a and b match. For 2 and 3 that is 2.4 against 2.5.
Why is the harmonic mean used for average speed?
Over the same distance you spend more time at the slower speed, so a plain average overstates the result. Equal distances at 40 and 60 km/h give 2 × 40 × 60 ÷ (40 + 60) = 48 km/h, not 50.
Does it work for more than two numbers?
The harmonic mean itself does — it is n divided by the sum of the reciprocals of n values. This calculator is built for exactly two inputs, which is why it uses the short form 2ab ÷ (a + b).
Sources and last check
Information, not professional advice.