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Harmonic Mean Calculator

Write down the values in the respective box, and the calculator will compute their harmonic mean.

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Harmonic Mean Calculator

Use this online Harmonic Mean Calculator to quickly calculate the harmonic mean of a dataset. It works for both positive and negative numbers and provides additional useful insights about your data.

What Is Harmonic Mean?

The harmonic mean (H.M.) is a measure of central tendency, alongside arithmetic and geometric means. It is the reciprocal of the arithmetic mean of reciprocals and generally produces the lowest value among the three means. It is sometimes called the subcontrary mean.

Harmonic Mean Formula

$$ H = \frac {n}{\frac {1}{x_1} + \frac {1}{x_2} + \dots + \frac {1}{x_n}} = \frac {n}{\sum_{i=1}^n \frac {1} {x_i}} $$

n = total number of values
x₁, x₂, …, xₙ = dataset values

Weighted Harmonic Mean

If weights ω₁, ω₂, …, ωₙ are associated with x₁, x₂, …, xₙ, the weighted harmonic mean is:

(Formula image placeholder)

Relation to Other Means

The harmonic mean is always the smallest among arithmetic and geometric means:

A.M > G.M > H.M

For two numbers, H.M can also be calculated as:

$$ H.M = \frac{G.M^2}{A.M} $$

Check our Geometric Mean Calculator for G.M. calculations.

How to Use the Harmonic Mean Calculator

Select the separator for your numbers (comma, space, or custom).
Enter your dataset.
Click Calculate.

Calculator Output

Harmonic mean
Numbers in ascending and descending order
Even and odd numbers
Total sum, maximum, minimum, and count of numbers

Applications of Harmonic Mean

Geometry: Radius of a triangle’s incircle = 1/3 of the harmonic mean of altitudes.
Finance: Price-to-earnings ratio across multiple items.
Physics: Average speed, electrical resistance, capacitance calculations.

Step-by-Step Example

Find the harmonic mean of 12, 23, 34, 45, 56:

n = 5
Data: x₁=12, x₂=23, x₃=34, x₄=45, x₅=56

Formula:

$$ H.M = \frac{5}{\frac{1}{12} + \frac{1}{23} + \frac{1}{34} + \frac{1}{45} + \frac{1}{56}} $$

Step-by-step:

$$ H.M = \frac{5}{0.083 + 0.043 + 0.029 + 0.022 + 0.017} $$

$$ H.M = \frac{5}{0.194} \approx 25.77 $$

FAQs

How do you calculate harmonic mean?

Take the reciprocal of each number, compute their arithmetic mean, then take the reciprocal of that result.

How do you calculate harmonic mean in Excel?

Use the HARMEAN function:

=HARMEAN(number1, [number2], ...)

How many harmonics are there?

Even harmonics
Odd harmonics

Merits and Demerits

Merits:

All values are considered.
Mathematically well-defined.
Supports algebraic manipulation.
Smoother curve than arithmetic or geometric means.

Demerits:

Harder for moderate mathematical knowledge to understand.
Complex calculations involving reciprocals.
Highly sensitive to extreme values.
Cannot be calculated if any value is zero.

Conclusion

The harmonic mean is widely used in finance, physics, geometry, and statistics. Use this online calculator to efficiently compute the harmonic mean for any dataset.

References

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