Harmonic Mean – Calculation & Uses ⚖️📐

The Harmonic Mean (H.M.) is a type of average most appropriate when the data are rates, ratios, or speeds, i.e., when the reciprocal of values is more meaningful than values themselves.

It is commonly used in finance, transport problems, and other situations where averages of rates are required.

Definition

Harmonic Mean of n observations X1, X2, ..., Xn is defined as the reciprocal of the arithmetic mean of reciprocals:

H.M. = n / (Σ (1 / Xi))

Example: For two numbers a and b,

H.M. = 2 / (1/a + 1/b) = 2ab / (a + b)

When Do We Use Harmonic Mean?

✔ For averaging rates or ratios

• Average speed over equal distances
• Cost per unit when units differ
• Price–earnings ratios across firms (under certain interpretations)

✔ When the reciprocal of values is additive

If the problem requires averaging 1/X, use harmonic mean.

Methods of Calculation

1. Direct Formula

H.M. = n / Σ(1/Xi)

Example: Values: 4, 6

H.M. = 2 / (1/4 + 1/6) = 2 / (0.25 + 0.1667) ≈ 2 / 0.4167 ≈ 4.8

2. Using Reciprocal Averages

Compute reciprocals Yi = 1/Xi, find arithmetic mean of Yi, then take reciprocal.

H.M. = 1 / ( (Σ Yi) / n )

3. Harmonic Mean for Grouped Data

When grouped by classes, use midpoints (X) as representative values and frequencies (f):

H.M. = (Σ f) / (Σ (f / X))

Example Table

Σf = 20
Σ(f/X) ≈ 0.7924
H.M. = 20 / 0.7924 ≈ 25.24

Properties of Harmonic Mean ⭐

1. Always ≤ Geometric Mean ≤ Arithmetic Mean

Equality holds only when all observations are equal.

2. Sensitive to small values

Because reciprocals of small numbers are large, a very small observation drags H.M. down significantly.

3. Useful for rates

Appropriate for averaging speeds, unit prices, and other rates.

4. Algebraic manipulability

H.M. is less convenient than A.M. for algebraic manipulation but still has useful relationships.

Advantages ✔️

• Correct for averaging rates (when weights are equal by distance or comparable base)
• Useful in certain financial ratios and speed problems

Limitations ❌

• Cannot be used if any value = 0 (division by zero)
• Heavily affected by very small values
• Less intuitive for students than A.M. and G.M.

Solved Examples

Example 1 — Average Speed over Equal Distances A car travels three equal-length segments at speeds 60 km/h, 80 km/h, and 120 km/h. Find the average speed.

Use H.M. because distances are equal:

H.M. = 3 / (1/60 + 1/80 + 1/120)
1/60 = 0.0166667
1/80 = 0.0125
1/120 = 0.0083333
Σ = 0.0375
H.M. = 3 / 0.0375 = 80 km/h

Interpretation: Average speed over equal distances = 80 km/h.

Example 2 — Grouped Data Example Given the midpoint-frequency table above, compute H.M. (worked earlier): result ≈ 25.24.

When to Choose Harmonic Mean vs Others

• Rates (speeds, time per unit) → Harmonic Mean
• Growth rates → Geometric Mean
• Additive quantities (sales, incomes) → Arithmetic Mean

Quick Flow for Students

Identify data type → Are values rates/ratios? → Any zeros? → Use H.M. if appropriate.

Summary ✨

• H.M. = n / Σ(1/Xi)
• Best for averaging rates and ratios when equal weighting is required.
• Sensitive to small values and cannot handle zeros.

Quiz Time 🎯