Harmonic Mean Calculator

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What is the harmonic mean?
When should I use the harmonic mean?
How to calculate the harmonic mean?
harmonic mean formula?
harmonic mean negative numbers
harmonic mean with zero
Example of harmonic mean

What is the harmonic mean?

The harmonic mean is a statistical measure of central tendency, representing the middle of a set of numbers. It calculates the arithmetic mean of the numbers' reciprocals (1/xi) and then reciprocates the result.

When should I use the harmonic mean?

You should use the harmonic mean when you need to average the duration of a rate problem.
The problem is a rate problem (units per duration), but you have duration data (duration per unit).
If the data would be rate data then you would use arithmetic mean (arithmetic).

We use the type of the average according to the question:
What is the average duration? Arithmetic Average.
What is the duration of the average pump? Harmonic Average.
The average pump is the pump that may replace all the pumps and get the same result.
See the following example.

How to calculate the harmonic mean?

1. Add up the reciprocal of all the numbers. ( Σreciprocals = 1/x1+1/x2+...+1/xn )
2. Count how many numbers there are. (n)
3. Divide the count by the addition. ( H = n / Σreciprocals )
1. Calculate the reciprocal of all the numbers. ( 1/x1,1/x2),...,1/xn )
2. Find the arithmetic average of the reciprocals. ( avg(reciprocals) = Σ(1/𝑥i)/n )
3. Calculate the reciprocal of the arithmetic average. ( H = 1 / avg(reciprocals) )

What is the harmonic mean formula?

H =n
1/x1 + 1/x2 + ... + 1/xn

Harmonic mean negative numbers

Usually, we use the harmonic mean with positive numbers.
The harmonic mean is a concave function, only when using positive numbers.
Example 1 Using two pumps, in opposite directions, one is filling the pool, while the second takes the water of the pool. Pump1 can fill a pool in 3 hours.
Pump2 can empty a pool in 4 hours.
Total Rate = 1/3 - 1/4 = 1/12.
It will take the two pumps 12 hours to fill the pool.
Using the two pumps is equivalent to using two pumps identical, that each can fill the pool in 24 hours.
H(3,-4) = 24.

Harmonic mean with zero

If it takes one pump to fill a pool for zero minutes, the rate is Infinity rate.
If you take the average of other rates with Infinity rate, the average rate will still be Infinity, Hence the duration to fill the pool will still be zero.

Example of the harmonic mean

Pump1 can fill a pool in 4 hours.
Pump2 can fill a pool in 8 hours.
The Harmonic mean answers the question: how long will it take for the average pump to fill the pool?

H =2
1/4 + 1/8

We may formulate the same question with rates
Rate =1 or  Duration =1
Pump1 can fill a 1/4 pool in one hour.
Pump2 can fill a 1/8 pool in one hour.
The arithmetic mean (arithmetic mean) answers the question: what is the rate of the average pump?
Average Rate =1/4 + 1/8