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Averages & Means

Weighted Harmonic Mean Calculator

Combine positive rates through exact weighted reciprocal averaging.

Averages & Means

Weight rates by exposure

Exact reciprocal weighting
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example
Paired rows

Use ordinary decimals; complete every field in each paired row.

Weighted harmonic mean

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Understand the subject

What is a weighted harmonic mean?

A weighted harmonic mean combines positive rates by adding the exposure that each rate covers. It is designed for situations where the rate sits in the denominator.

The relationship

The defining relationship

The positive xi are rates; wi represent matching numerator-side exposure, such as distance when averaging speed.

See the idea

Equal distance means add times first

Worked example

Two equal-distance travel legs

At 60 km/h and 40 km/h over equal distances, the average speed is 2 ÷ (1/60 + 1/40) = 48 km/h, not 50 km/h.

Interpret with care

Choose the relationship that fits

The weights must reflect the right exposure. For speeds observed for equal time instead, the arithmetic mean is the relevant relationship.

For a side-by-side view of the classical means, use the Mean Comparison Calculator.

Quick guide

How to use this calculator

  1. Enter a positive rate or ratio and its matching non-negative weight.
  2. Choose weights that represent the appropriate numerator-side exposure.
  3. Inspect the exact reciprocal sum and total weight.

Calculation method

Weight reciprocals, then invert

The result is total weight divided by the sum of each weight divided by its matching value.

Worked example

Combine unequal rate weights

Rates 2 and 4 with equal weights have weighted harmonic mean 8/3.

Hᵥ = (1+1)/(1/2+1/4) = 8/3

Supported inputs

Precision and limits

List size

Add 1–100 labelled value rows.

Decimal size

Each value may contain up to 200 digits and 100 decimal places.

Exactness

Exact rational results are never replaced by a rounded decimal; approximations are labelled.

Domain

Values must be positive. Zero weights are ignored mathematically; total weight must remain positive.

Interpretation

Correct weight selection depends on what the rate's numerator and denominator represent.