When should I use the harmonic mean?

Use it for rates, ratios, and per-unit quantities. It emphasizes smaller values, making it ideal for averaging speeds, price ratios, or similar metrics.

Can values be zero or negative?

Zeros are not allowed. Negative values are mathematically permitted but typically avoided in practical scenarios where quantities are inherently positive.

How does it differ from the arithmetic mean?

Harmonic mean averages reciprocals and thus emphasizes smaller values, while the arithmetic mean averages the values directly.

What if weights don’t sum to one?

Normalization isn’t required. The formula uses the raw sum of weights over the sum of weighted reciprocals.

How many decimal places should I choose?

Pick more decimals for precision-focused work; fewer decimals for readability. This tool supports 0–10 decimal places.

Does unit consistency matter?

Yes, ensure all inputs use compatible units. The resulting unit matches the input values.

Why is the harmonic mean ≤ arithmetic mean?

For positive values, AM–HM inequality guarantees the harmonic mean is not greater than the arithmetic mean.

Harmonic Mean Calculator

Input

White Theme

Values (separate by comma, space, or newline)

Examples: 40 50 60 72 or 40,50,60,72

Please enter at least one value.

Weights (optional; same count as values)

Leave empty for unweighted mean. Weights must be nonnegative.

Mode

Unweighted

Weighted

Decimal places

Example Data Table

Click “Use This Example” to populate the form with this dataset.

#	Value	Weight	Context
1	40	1	Rate sample
2	50	2	Rate sample
3	60	3	Rate sample
4	72	2	Rate sample

Harmonic Mean Reference Data — Unweighted

Precomputed examples. Click Use to load values into the calculator.

Dataset	Values	Harmonic Mean
Equal-distance speeds (km/h)	40, 50, 60, 72	52.9412
Resistances (Ω)	10, 20, 30	16.3636
Price per unit ($/unit)	1.2, 1.5, 1.4	1.3548

Harmonic Mean Reference Data — Weighted

Weighted examples with nonnegative weights. Click Use to load both values and weights.

Dataset	Values	Weights	Weighted HM	Action
Rates with observation counts	40, 50, 60, 72	1, 2, 3, 2	56.0311
Mixed-quality samples	20, 30, 40, 50	5, 3, 2, 1	26.1905

Results

Mode	Unweighted
Input values	40, 50, 60, 72
Weights
n	0
∑(1/x)
∑w
∑(w/x)
Harmonic Mean

Note: Values must be nonzero. Weighted mode requires weights count to match.

Formula Used

Unweighted: For nonzero values x₁, x₂, …, x_n, the harmonic mean is

H = n / [ (1/x₁) + (1/x₂) + … + (1/x_n) ]

Weighted: For positive weights w_i,

H_w = (∑w_i) / (∑ (w_i / x_i))

Zeros are not allowed since division by zero is undefined.
Harmonic mean is suitable for rates and ratios (e.g., speeds).
Units follow the value units (e.g., km/h, Ω when using resistances in parallel contexts).

Interactive Charts

Show reciprocals

Tip: use the camera icon on each chart to download as PNG.

How to Use This Calculator

Paste or type values separated by comma, space, or newline.
Optionally enter weights matching the number of values.
Select Unweighted or Weighted mode.
Choose decimal places and press Compute.
Review results and intermediate sums for verification.
Export results as CSV or generate a PDF report.
Use the example table to experiment and learn quickly.

Frequently Asked Questions

Use it for rates, ratios, and quantities expressed as “per unit,” such as speeds over fixed distances or average price ratios. It emphasizes smaller values appropriately.

Zeros are not allowed because 1/0 is undefined. Negative numbers are mathematically permitted, but interpret results carefully; most practical applications expect positive values.

The arithmetic mean averages values directly, while the harmonic mean averages reciprocals. Harmonic mean gives more weight to smaller values, making it ideal for averaging rates.

No normalization is necessary. The formula divides the sum of weights by the sum of weighted reciprocals, so any nonnegative weights work as long as their sum is nonzero.

Use more decimals for financial or scientific precision. For quick comparisons, fewer decimals keep results readable. This tool supports 0–10 decimal places.

Yes. Keep all values in compatible units before computing. The result inherits the same unit as the original values (e.g., km/h, Ω, $/unit).

By AM–HM inequality, for positive values the harmonic mean never exceeds the arithmetic mean, reflecting its sensitivity to smaller numbers in a dataset.