HARMEAN Function in Apple Numbers: Harmonic Mean and Rate Examples

=HARMEAN(pos-num, pos-num…)

Apple’s documentation names number values, not collections, for these arguments. Its generic Types of arguments and values defines number value to include an expression or cell containing a number, while a collection is a separate range/array type. Accordingly, this guide uses distinct cells, such as B2,B3, instead of assuming HARMEAN(B2:B3) must be supported merely because it works in another application.

No weights, default substitute for missing optional arguments, zero-handling flag or special unit conversion appears in the documented syntax. A missing optional argument is simply omitted. Zero is not valid because the reciprocal is undefined; negative values also fail Apple’s positive-input definition.

Create an Apple Numbers table named Trips. Enter these exact rows and make B a numeric column representing kilometres per hour. Put the result in D2 in the same table to avoid an unqualified cross-table reference.

=HARMEAN(B2,B3)

Independently calculated expected answer: 40 km/h. For equal 60-kilometre segments, the slower segment takes 2 hours and the faster takes 1 hour. Total travel is 120 km over 3 hours, so 120 ÷ 3 = 40 km/h. The harmonic-mean calculation agrees: 2 ÷ (1/30 + 1/60) = 40.

For contrast:

=AVERAGE(B2,B3)

The arithmetic result is 45, but 45 km/h would incorrectly assume equal time spent at each speed. AVERAGE is appropriate when observations have equal time weights (or when average speeds are being summarized as separate observations), not for the physical distance/time of two equal-distance segments.

Why the setup matters: If the distances differ, an unweighted harmonic mean is not automatically the trip average. Calculate total distance divided by total travel time using the actual segment lengths. This illustrates the correct scope of the formula, rather than promising a universal rate-averaging shortcut.

Replace the speed column with 10, 20, and 40 in B2, B3 and B4, adding a third segment row. Then:

=HARMEAN(B2,B3,B4)

Independent expected answer: 120/7, approximately 17.1428571429. The sum of reciprocals is 1/10 + 1/20 + 1/40 = 7/40; dividing three by 7/40 gives 120/7.

For comparison, =AVERAGE(B2,B3,B4) has expected result 23.3333333333. Rates that are lower influence a harmonic mean more strongly, which is precisely why this mean is useful for repeated equal-distance travel and certain reciprocal relationships. Both calculations presume each observation has the role described; do not apply the rate interpretation to counts of unrelated events.

Audit the formula without HARMEAN

With three positive values the equivalent arithmetic expression is:

=3/(1/B2+1/B3+1/B4)

Expected output: the same 120/7. This gives a transparent cross-check of the reciprocal method and is independently calculated. It is not native proof that HARMEAN produces exactly this rendered decimal in your version of Numbers.

Note about Apple’s published decimal example

The official article demonstrates =HARMEAN(1,2,8) and explicitly gives the exact fraction 24/13. Our independent fraction calculation also gives 24/13 ≈ 1.846153846153846….

However, at the accessed date the article also printed 1.84615386415385, which differs from 24/13 in the last digits. This is a documented inconsistency in the displayed Apple example, not evidence of a different harmonic-mean formula. In the tutorial, the exact fraction controls the independently calculated result. A native test would be required to see what a particular Numbers version actually displays. No undocumented error-handling or rounding rule is inferred from this discrepancy.

• Zero in any position: Not permitted by the >0 argument restriction and makes reciprocal division impossible. Check source measurements; do not treat zero as a missing value without a documented data-handling policy.
• Negative numbers: The current official HARMEAN signature requires positive terms. A negative rate is not a valid drop-in input; resolve the meaning of the sign separately.
• Blanks, text, Booleans, durations: The source page documents positive number values; it does not establish silent conversion or omission rules for arbitrary mixed-range cells. Do not borrow another spreadsheet’s coercion assumptions.
• The range shortcut does not work: Because the official argument is pos-num (number value), list individual numeric cells first and verify a multi-cell collection explicitly before adopting one.
• Unexpectedly low result: The harmonic mean puts relatively greater weight on smaller rates. Check that all speeds are expressed in matching units and correspond to equal distances if applying the trip interpretation.
• Wrong reference table: When a result is outside Trips, Numbers needs a table-qualified reference. Use the formula editor to select the intended source cells.
• Formula error after regional settings change: Use semicolons instead of commas between arguments when your decimal separator is a comma. Decimal display and separators are different settings.
• Floating-point differences: The exact harmonic mean may be nonterminating in decimal. Round for display, but keep full precision for subsequent calculations. Apple’s help does not promise an exact decimal rendering.
• Insufficient or malformed input: The first pos-num argument is mandatory. Verify argument count and parentheses instead of relying on an assumed fallback result.

These programs also use the harmonic-mean concept for positive data, but application-specific details about passing ranges, ignoring cells, coercing text, displaying zero errors, and separators are not certified here. Do not copy a working Excel HARMEAN(A2:A9) into Numbers and assume its collection behaviour or reference names carry over. Apple’s own argument-type description, rather than another spreadsheet’s syntax, controls this page.

• AVERAGE — arithmetic mean for additive observations and equal-duration speed measurements.
• MEDIAN — the middle ranked number, not a reciprocal mean.
• PRODUCT — multiplication; relevant when deriving a geometric mean, not a replacement for HARMEAN.

This page was researched using the Apple Formulas and Functions Help, edition 15.4, accessed 2026-10-12 UTC. This edition does not by itself prove compatibility with every historical Numbers release or every platform. Not executed in Apple Numbers on macOS, iPhone, iPad, or iCloud.com. The displayed answers below are mathematical or documentation-checked expected results, not screenshots or observed output.

The English help examples separate arguments with commas. Apple documents that regions using a comma decimal separator use a semicolon between arguments typed by hand: see Functions overview. Use the Numbers formula editor to construct references. In another table Numbers uses qualifiers such as Data::B2, not Excel-style Sheet1!B2; see Apple’s argument types and table-reference guidance. Regional number displays can differ without changing the underlying mathematics.

The first-supported Numbers version and older-platform differences were not verified for this function. Do not assume a version simply because the current documentation is edition 15.4.

Official claim sources, checked 2026-10-12 UTC:

• Apple HARMEAN function help, documented as edition 15.4 and accessible Apple function article: function signature, input requirements, and function-specific notes.
• Apple Functions overview: syntax notation and localized argument separators.
• Apple Types of arguments and values: number/collection distinction and Numbers table-qualified references.

Validation: Official documentation and independent arithmetic only; no native Numbers execution or screenshots.