About

The Harmonic Mean is one of the three Pythagorean means (alongside Arithmetic and Geometric). It is specifically designed for situations where the average of rates or ratios is required. For example, if a vehicle travels a set distance at speed A and returns at speed B, the average speed is not the arithmetic mean (A+B)/2, but the harmonic mean.

This tool is essential for physicists, engineers, and financial analysts dealing with P/E ratios or cost averaging. It features a robust parser that allows you to paste messy data (e.g., "10km, 20km, 30km") and instantly extracts the valid numerical values for calculation.

Formulas

For a set of n numbers x₁, x₂, ..., x_n, the Harmonic Mean H is defined as the reciprocal of the arithmetic mean of the reciprocals:

H = nn∑i=1 1x_i

Note: The Harmonic Mean is undefined if any x is zero, and typically used for positive real numbers.

Reference Data

Scenario	Values (x)	Arithmetic Mean	Harmonic Mean (H)	Why Harmonic?
Speeds (Fixed Distance)	60, 40 km/h	50	48	Time spent at lower speed is longer.
Resistors (Parallel)	100, 100 Ω	100	100 (Total R = H/n = 50)	Inverse relationship of conductance.
Finance (P/E Ratios)	10, 20	15	13.33	Price is numerator, Earnings denominator.
Data Efficiency	2, 4, 8	4.66	3.43	Mitigates impact of large outliers.

Frequently Asked Questions

Use the Harmonic Mean when you are averaging rates where the total amount of work (or distance) is fixed, but the time or resources taken varies. Common examples include speed (distance is fixed), parallel resistance (voltage is fixed), or capacitors in series.

Mathematically, the formula works with negative numbers, but the result may not be meaningful in physical contexts like speed or resistance. Mixed positive and negative numbers can result in division by zero if the sum of reciprocals is zero.

This is a property known as the inequality of arithmetic and harmonic means. The Harmonic Mean is more heavily influenced by smaller numbers in the dataset. If all numbers are equal, the means are equal; otherwise, Harmonic ≤ Geometric ≤ Arithmetic.