Harmonic Mean is the reciprocal of the arithmetic mean of reciprocals; used for averaging rates, speeds, and ratios where the harmonic relationship matters.
## Core concept
The Harmonic Mean (HM) is a measure of central tendency specifically designed for data that involves rates and ratios. Unlike arithmetic mean which sums values directly, harmonic mean emphasizes smaller values and is appropriate when the average of reciprocals is meaningful.
When to use HM: - Average speed over a journey (distance constant, time varies) - Average price per unit when total amount spent is fixed - Average cost per item - Rates of work, production rates - Pulse rates, frequency-based data
Why not AM? If you travel 100 km at 50 km/h and 100 km at 100 km/h, the average speed is NOT 75 km/h (AM of speeds). It's the HM because distance is held constant.
## Formula / rule
For ungrouped data (n observations):
$$HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}} = \frac{n}{\sum \frac{1}{x_i}}$$
For grouped/frequency data:
$$HM = \frac{\sum f}{\sum \left(\frac{f}{x}\right)}$$
where f = frequency, x = class midpoint (or value)
Relationship with AM and GM: - AM ≥ GM ≥ HM (always, for positive values) - Equality holds only when all values are identical - For two numbers a and b: $GM^2 = AM \times HM$
## Common exam applications
Example 1: Average Speed A car travels 120 km at 60 km/h and 120 km at 40 km/h. Find average speed.
$$HM = \frac{2}{\frac{1}{60} + \frac{1}{40}} = \frac{2}{\frac{2+3}{120}} = \frac{2 \times 120}{5} = 48 \text{ km/h}$$
(Check: Total distance = 240 km; Total time = 2 + 3 = 5 hours; Average = 240/5 = 48 km/h ✓)
Example 2: Average Price per Unit ₹1000 spent on apples at ₹50/kg and ₹1000 on apples at ₹25/kg. Average price?
Quantity at ₹50 = 20 kg; Quantity at ₹25 = 40 kg $$HM = \frac{2000}{\frac{1000}{50} + \frac{1000}{25}} = \frac{2000}{20 + 40} = \frac{2000}{60} = ₹33.33 \text{ per kg}$$
## Common mistakes
- Using AM when HM is required: Leads to incorrect average speed, price, or rate. The error compounds when values differ widely.
- Forgetting the constraint: HM applies only when one factor is constant (distance, money spent, output quantity). If time itself varies in a speed problem, use AM.
- Reciprocal errors: Ensure you're adding reciprocals correctly; careless arithmetic is common.
- Zero or negative values: HM is undefined for zero or negative values. Check data validity.
- Not matching the structure: In "₹X spent" problems, always use HM. In "₹X per unit" problems with equal quantities, use AM.
## Quick verification checklist
✓ Does the problem involve rates or ratios? ✓ Is one component held constant (distance, expenditure, output)? ✓ Are all values positive? ✓ Have you computed $\sum \frac{1}{x_i}$ correctly?
Use this alongside AM, GM, and median questions in your practice sets. HM appears 1–2 times per foundation exam, often as an application problem rather than pure definition.