Harmonic Mean Calculator
Calculate harmonic mean ($\text{HM} = \frac{N}{\sum 1/x}$), average speed across equal distances, unit rate averages, and price-to-earnings (P/E) ratios.
Calculate Harmonic Mean & Rate Averages
Enter positive numbers (> 0) separated by commas, spaces, or lines.
Calculation Results
Calculated using harmonic reciprocal mean equations: \text{HM} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}}
Quick Summary
The Harmonic Mean Calculator evaluates the reciprocal average ($\text{HM} = \frac{N}{\sum 1/x}$), useful for average speeds over fixed distances and financial valuation ratios (P/E).
Formula Explanation
\text{Harmonic Mean HM} = \frac{N}{\sum_{i=1}^{N} \frac{1}{x_i}} = \frac{N}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_N}}How It Works
The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. The Harmonic Mean Calculator averages rates expressed as ratios (such as miles per hour or price per earnings) where the numerator dimension is held constant.
Step-by-Step Worked Example
Practical Problem: A car travels a 100-mile trip at 60 mph going out, and returns over the exact same 100-mile distance at 40 mph coming back. Calculate average speed for the round trip.
- Step 1: Calculate reciprocals of speeds: $1/60 = 0.016667$; $1/40 = 0.025000$.
- Step 2: Sum the reciprocals ($\sum 1/x$): $0.016667 + 0.025000 = \mathbf{0.041667}$.
- Step 3: Divide total count ($N = 2$) by sum of reciprocals: $\text{HM} = 2 / 0.041667 = \mathbf{48.0000\text{ mph}}$.
- Step 4: Compare with misleading Arithmetic Mean: $(60 + 40) / 2 = \mathbf{50.00\text{ mph (Incorrect!)thrust}}$.
- Step 5: Interpretation: The true average speed over equal distance is 48.00 mph (since traveling at 40 mph takes more time than 60 mph).
Real-World Calculation Examples
Scenario 1: Round Trip Speed (60 mph & 40 mph)
Parameters: 60 mph, 40 mph equal distance
Result: 48.00 mph Harmonic Mean Speed.
Scenario 2: Financial Stock P/E Ratio Average
Parameters: Stock P/E 10, Stock P/E 20
Result: 13.33 Average P/E Ratio (Harmonic mean avoids upward bias).
Scenario 3: Machine Work Output Rates
Parameters: 10 units/hr, 20 units/hr, 30 units/hr
Result: 16.36 units/hr Harmonic Mean Rate.
Scenario 4: Parallel Electrical Resistors (R1, R2)
Parameters: 100 ohms, 100 ohms
Result: 50.00 ohms Parallel Resistance ($\text{HM} / 2$).
Key Benefits of Using This Calculator
Correct Speed & Rate Averaging
Computes true average speed over fixed distances without time-weighting errors.
Unbiased Stock Portfolio P/E Ratios
Averages financial price-to-earnings (P/E) ratios without overstating expensive valuation multiples.
Pythagorean Means Comparison
Displays Harmonic Mean alongside Arithmetic Mean and Geometric Mean.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is harmonic mean?
Harmonic mean is a Pythagorean average calculated by dividing observation count $N$ by the sum of reciprocals ($\frac{N}{\sum 1/x}$).
Why is harmonic mean used for average speed?
When traveling fixed distances at different speeds, lower speeds take longer time, weighting total elapsed time. Harmonic mean calculates true overall speed.
Can harmonic mean be calculated with zero or negative numbers?
No — reciprocal $1/0$ is undefined (division by zero), so harmonic mean requires strictly positive numbers ($>0$).
Why is harmonic mean used for financial P/E ratios?
P/E is price per share divided by earnings per share. Averaging P/E ratios directly gives excessive weight to high P/E stocks; harmonic mean averages E/P earnings yields correctly.
What is F1-Score in machine learning evaluation?
F1-score is the harmonic mean of precision and recall ($\text{F1} = \frac{2 \cdot \text{Precision} \cdot \text{Recall}}{\text{Precision} + \text{Recall}}$).
How does harmonic mean compare to arithmetic and geometric mean?
Pythagorean Inequality: $\text{Harmonic Mean} \le \text{Geometric Mean} \le \text{Arithmetic Mean}$.
What is weighted harmonic mean?
Weighted harmonic mean is $\text{HM}_w = \frac{\sum w}{\sum (w / x)}$.
How do I calculate harmonic mean in Excel?
Use Excel formula =HARMEAN(range).
How is harmonic mean used in electrical engineering?
Harmonic mean calculates total equivalent resistance of parallel resistors ($R_{\text{eq}} = \frac{1}{\sum 1/R_i}$).
When should I NOT use harmonic mean?
Do not use harmonic mean when rates are measured over equal time intervals (use standard arithmetic mean for equal time intervals!).