Quadratic Equation Solver (ax² + bx + c = 0)
About Harmonic Mean Calculator
The Harmonic Mean ($HM$) & Rate-Averaging Calculator computes the harmonic mean ($HM = n / \sum (1/x_i)$) for averaging ratios, speeds (km/h, mph), financial Price-to-Earnings ($P/E$) ratios, fuel efficiency (MPG), and machine-learning F1-scores.
How to Use Harmonic Mean Calculator
Step 1
Enter positive rate/speed numbers separated by commas or spaces.
Step 2
Review computed harmonic mean and reciprocal sum breakdown.
Step 3
Inspect comparisons with Arithmetic and Geometric means.
Step 4
Click "Copy Result".
Practical Use Cases for Harmonic Mean Calculator
Calculating True Average Round-Trip Travel Speeds (mph / km/h)
Calculate correct average velocity when driving 60 mph outbound and 30 mph return over the same fixed-distance route ($HM = 40\text{ mph}$, not 45 mph).
Machine Learning Precision-Recall F1-Score & Financial P/E Valuation
Compute balanced harmonic means between precision and recall ($F_1 = 2 \times (P \times R)/(P + R)$) and portfolio valuation multiples.
Input & Output Examples
Averaging Round-Trip Driving Speeds (60 mph and 30 mph)
Speeds: `60, 30`
Harmonic Mean: `40.0 mph` (Correct Average Speed) | Arithmetic Mean: `45.0 mph (Flawed Average)`
Key Features & Performance
- ✓Computes: Harmonic Mean ($HM$), Weighted Harmonic Mean, and compares with Arithmetic and Geometric Means.
- ✓Specialized modes: Average Speed over Distance, Fuel Economy (MPG / L/100km), and ML $F_1$ Score.
- ✓Step-by-step reciprocal summation proofs.
- ✓100% Client-Side memory execution.
- ✓1-Click Copy harmonic mean calculation.
Key Terminology & Definitions
Harmonic Mean ($HM$)
The reciprocal of the arithmetic mean of the reciprocals of a given set of positive numbers.
Rate Averaging
When averaging quantities defined as ratios with equal numerators (e.g. kilometers per hour where distance is constant), the harmonic mean yields the true physical average.