Arithmetic, geometric and harmonic mean

There are three classical means, and choosing between them is not a matter of preference: each answers a different kind of question. Using the arithmetic mean where the geometric belongs is one of the most common errors in statistics — and one of the most distorting in investment reporting.

The three formulas

• Arithmetic: (a + b) ÷ 2 — add and divide.
• Geometric: √(a × b) — multiply and take the root. For n values, the nth root of the product.
• Harmonic: 2 ÷ (1/a + 1/b) — the mean of the reciprocals, inverted.

They never give the same answer

For the numbers 4 and 9:

Mean	Calculation	Result
Arithmetic	(4 + 9) ÷ 2	6.5
Geometric	√(4 × 9) = √36	6.0
Harmonic	2 ÷ (1/4 + 1/9)	5.54

The order never changes: arithmetic ≥ geometric ≥ harmonic, with equality only when every value is identical. The more spread out the numbers, the wider the gaps.

When to use each

Arithmetic — quantities that add up. Grades, heights, temperatures, salaries. If it makes sense to ask "what would the total be if all were equal", this is the one.

Geometric — quantities that multiply. Growth rates, investment returns, accumulated inflation, population change. Whenever values chain together by multiplication, this is the one.

Harmonic — ratios with a fixed numerator. Speeds over the same distance, average price per unit when spending a fixed amount, portfolio-level ratios like P/E.

The expensive error: average return

An investment gains 100% one year and loses 50% the next. The arithmetic mean of those returns is +25% a year — a figure that turns up regularly in marketing material.

The real result is zero: $100 becomes $200 and returns to $100. The geometric mean gives exactly that: √(2.00 × 0.50) = 1.00, or 0% a year.

The same holds with less extreme numbers: a 50% gain followed by a 33.33% loss gives an arithmetic mean of +8.33% and a geometric mean of 0% — which is the actual return. That is why the correct figure for describing performance over time is the geometric mean, also called the annualised return or CAGR.

The travel error: average speed

You drive a route at 40 mph and return the same way at 60 mph. What was the average speed?

Not 50 mph. It is 48 mph — the harmonic mean. The reason is intuitive once said out loud: you spent more time going slowly. The 40 mph leg occupies more of the trip's hours than the 60 mph leg, so it weighs more.

The check is direct. Over 120 miles each way: 3 hours out, 2 hours back, 240 miles in 5 hours = 48 mph.

The general rule: when the quantity is a ratio and the numerator is what stays fixed (the distance, here), the harmonic mean is the correct one.

A three-question test

When unsure which to use, answer in this order:

1. Do the values add up to a meaningful total? If so, arithmetic. Summing grades, salaries or heights produces an interpretable total.
2. Do the values multiply in a chain? If so, geometric. Growth of 10% followed by 20% is not 30%, it is 1.10 × 1.20 = 32%.
3. Are the values ratios with the same numerator? If so, harmonic. Miles per hour over the same distance, dollars per unit over the same amount spent.

If none of the three fits, the question probably does not want a mean at all but the whole distribution — median, quartiles, minimum and maximum.

Frequently asked questions

Does the geometric mean work with negative numbers? Not directly — the root of a negative product is not real. So with returns you work with factors (1 + return): a 20% loss becomes 0.80, not −0.20.

Which mean gives my average share price when buying over time? The quantity-weighted one, which is an arithmetic mean with weights. But if you invested the same dollar amount at each purchase, your average price paid is the harmonic mean of the prices, and it is lower than the arithmetic one. That is the mathematical reason dollar-cost averaging buys more shares when prices fall.

Where does the median fit in? In a different category: the three means summarise magnitude, the median summarises position. When extreme values are present, the median usually describes the typical case better than any of the three.

Do they work for three or more values? Yes, all of them generalise: the geometric mean of n values is the nth root of the product, and the harmonic is n divided by the sum of the reciprocals.