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Mean Calculator (Arithmetic, Geometric, Harmonic, Quadratic)

Computes the Pythagorean means (arithmetic, geometric, harmonic) of a list of numbers together with the quadratic mean (root mean square), the contraharmonic mean, the midrange and the median.

When to use

You need a specific kind of average, e.g. the geometric mean of growth factors, the harmonic mean of rates or speeds, or the RMS of signal values, or want to compare the different means of one data set.

Do not use when: You need dispersion measures such as variance and standard deviation (use descriptive-statistics) or a weighted average (use weighted-average).

Formula

A = Σx/n; G = (Πx)^(1/n); H = n/Σ(1/x); Q = √(Σx²/n); C = Σx²/Σx; midrange = (min + max)/2

Inputs

Parameter	Type	Unit	Required	Description
`numbers`	number_list		yes	The values, comma-separated. Geometric and harmonic means require all values to be positive.

Outputs

Output	Type	Description
`arithmetic_mean`	number	Σx / n.
`geometric_mean`	number	(Πx)^(1/n), computed as exp(mean(ln x)); only for positive values.
`harmonic_mean`	number	n / Σ(1/x); only for positive values.
`quadratic_mean`	number	√(Σx² / n), the root mean square.
`contraharmonic_mean`	number	Σx² / Σx (requires Σx ≠ 0).
`midrange`	number	(min + max) / 2.
`median`	number	Middle value of the sorted list (mean of the two middle values for an even count).
`count`	integer	Number of values n.
`inequality`	string	For positive values: harmonic ≤ geometric ≤ arithmetic ≤ quadratic ≤ contraharmonic, with equality only when all values are equal.

Example

1, 2, 4, 8: {"numbers":[1,2,4,8]} → {"arithmetic_mean":3.75,"geometric_mean":2.828427,"harmonic_mean":2.133333,"quadratic_mean":4.609772,"contraharmonic_mean":5.666667,"midrange":4.5,"median":3,"count":4}

4 and 9: {"numbers":[4,9]} → {"arithmetic_mean":6.5,"geometric_mean":6,"harmonic_mean":5.538462,"quadratic_mean":6.964194,"contraharmonic_mean":7.461538,"midrange":6.5,"median":6.5,"count":2}

GET https://tttkmbb.com/api/v1/calculate/mean-types?numbers=1%2C2%2C4%2C8

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/mean-types (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/mean-types · Markdown: https://tttkmbb.com/math/mean-types.md · JSON definition: https://tttkmbb.com/math/mean-types.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="mean-types"
OpenAPI operationId: calculate_mean_types
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Pythagorean means (reference)
Wikipedia – Root mean square (reference)
Wikipedia – Contraharmonic mean (reference)

FAQ

When should I use the geometric or harmonic mean?

Geometric for quantities that multiply (growth rates, ratios, index numbers); harmonic for rates over a fixed quantity (average speed over equal distances, price/earnings ratios). The arithmetic mean overstates both.

Why is the geometric mean missing?

It is only defined for positive values (a zero makes it 0 and negatives make it complex), so it and the harmonic mean are omitted, with a note, when the list contains zero or negative numbers.

Related calculators

Descriptive Statistics Calculator — Full summary statistics including variance and quartiles.
Weighted Average Calculator — Average with different weights per value.
CAGR Calculator — The geometric mean growth rate of an investment.