Harmonic Mean Calculator

Use this unified calculator to find the harmonic mean for both ungrouped (raw) data and grouped (frequency distribution) data. The harmonic mean is ideal for averaging rates and ratios.

Quick Start

Harmonic Mean Calculator
Data Type	Ungrouped (Raw Data) Grouped (Frequency Distribution)
Enter the X Values (Separated by comma,)
Type of Frequency Distribution	DiscreteContinuous
Enter the Classes for X (Separated by comma,)
Enter the frequencies (f) (Separated by comma,)

Results
Number of Observations (n):
Harmonic Mean: (HM)

Understanding Harmonic Mean

Harmonic Mean is used for:

Rates and speeds - average speed over a journey
Ratios - price per share, cost per unit
Reciprocal relationships - frequency, density
Cannot be used for zero or negative values
Most conservative of the three means (AM ≥ GM ≥ HM)

Formulas

Ungrouped Data: $$HM = \frac{n}{\sum_{i=1}^{n}\frac{1}{x_i}}$$

Grouped Data: $$HM = \frac{N}{\sum_{i=1}^{n}\frac{f_i}{x_i}}$$

Where:

n = number of observations (ungrouped)
N = total frequency (grouped)
xᵢ = observation or class value
fᵢ = frequency

Alternative Form

$$\frac{1}{HM} = \frac{1}{n}\sum_{i=1}^{n}\frac{1}{x_i}$$

When to Use Harmonic Mean

Scenario	Example	Formula
Average speed	Drove 60 mph for 2 hours, then 40 mph for 2 hours	HM of 60, 40
Average price per share	First buy: 10 shares @ $50, Second buy: 10 shares @ $100	HM of prices
Ratios	Speed, productivity rates, efficiency	Use HM of rates

Classic Example:

Trip A: 100 km at 60 km/h
Trip B: 100 km at 40 km/h
Average speed = HM(60, 40) = 48 km/h (NOT the arithmetic mean of 50 km/h)

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