Head-to-head
Geometric Mean vs Weighted Average
An average is a summary, and which summary you pick decides what the number means. The arithmetic mean adds the values and divides by how many there are. The geometric mean multiplies them and takes a root. The weighted average multiplies each value by a weight before adding. On three years of returns these three produce three different answers, and only one of them describes what happened to the money.
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The geometric mean is the right summary for anything that compounds, because compounding is multiplication. Two years of +50% and −50% leave you with 75% of what you started with, not 100%. The arithmetic mean says the average year was flat; the account says otherwise. The geometric mean of those two factors is 0.866, or −13.4% a year, which is exactly what the balance did.
A weighted average answers a different question: given values that count for different amounts, what single equivalent value applies? A course grade with a final worth 60% and coursework worth 40% is the standard case. The weights are a statement about importance, not about time, so a weighted average is perfectly usable on values that never compound at all.
The confusion starts when a weighted average is applied to returns with equal weights and then read as a growth rate. With equal weights it collapses into the arithmetic mean, and the arithmetic mean overstates compound growth by roughly half the variance. That is why a volatile portfolio's quoted average return is always a little better than its actual return.
Geometric mean
How To Calculate A Geometric Mean
The geometric mean is the average you use when values multiply rather than add. Multiply all the numbers together and take the root that matches how many there are.
Weighted average
How To Calculate A Weighted Average
A weighted average answers "what is the typical value?" when some values count more than others. Grades, portfolio returns, inventory costs and survey results are almost never plain averages, which is why the distinction matters.
How They Differ
| Attribute | Geometric mean | Weighted average |
|---|---|---|
| What it does | Multiplies the values, takes the nth root | Multiplies each by a weight, then adds |
| Right for | Rates that compound over time | Values of unequal importance |
| Wrong for | A weighted grade, where the years do not compound | A growth rate, which it overstates |
| With equal weights | Stays the geometric mean | Collapses into the arithmetic mean |
| Three years: +50%, −50%, +20% | −3.45% a year | +4.00% at weights of 40, 40 and 20 |
| What it implies for $10,000 | $9,000 — which is what happened | $11,249 if compounded, which did not |
| Reaction to a single zero | The whole product becomes zero | Unaffected |
| Units | Same as the values, when they are factors | Same as the values |
The Same Numbers, Both Ways
Three years of returns — +50%, −50% and +20% — summarised once by the geometric mean and once by a weighted average using weights of 40%, 40% and 20%.
- Year 1
- +50%
- Year 2
- −50%
- Year 3
- +20%
- Weights
- 40%, 40%, 20%
| Line | Geometric mean | Weighted average |
|---|---|---|
| Summary of the three years | -3.45% | 4.00% |
| Equal-weight arithmetic mean | Not used | 6.67% |
| Three factors multiplied | 0.9000 | 0.9000 |
| $10,000 after the three years | $9,000 | $9,000 |
| If the summary rate were compounded | $9,000 | $11,249 |
| Uses the weights | No — every year counts once | Yes: 40%, 40%, 20% |
| Correct as a growth rate | Yes | No — it overstates compounding |
The three years multiply to 0.9000 of the starting value, so the geometric mean is -3.45% a year and $10,000 becomes $9,000. The weighted average of the same three numbers is 4.00%, and compounding that for three years would claim $11,249 — $2,249 that the account never earned.
When To Use Each
Geometric mean
- You are summarising returns, growth rates, inflation or anything else that compounds period over period.
- You want the single rate that turns the starting value into the ending value.
- You are comparing two volatile series and need a figure that does not reward volatility.
Weighted average
- The values differ in importance rather than in time — grades, survey strata, index components.
- You are computing a portfolio return for a single period, where the weights are the holdings.
- The values do not compound at all, such as marks or prices in one period.
Common Questions
Why not just use the arithmetic mean?
Because it answers a question nobody asks. It is the mean of the yearly factors, not the rate that turns the starting balance into the ending one. The gap grows with volatility, so the more the returns swing, the more the arithmetic mean flatters them.
When is a weighted average the right choice?
When the values differ in importance rather than in time — grades, index components, holdings, survey strata. If the weights sum to one and the values are rates for the same period, the weighted average is the portfolio return for that period.
Is there a weighted geometric mean?
Yes, and it is the correct summary when compounding and unequal weights both apply, such as a portfolio of assets each growing at its own rate. Most spreadsheets do not offer it, which is why it is often approximated badly or skipped altogether.
Does the geometric mean work with negative returns?
Yes, if you convert them to factors first: −50% becomes 0.5, and the product stays positive as long as no year returns −100%. A total loss makes the product zero and the geometric mean zero, which is the correct answer.
Why does one zero break the geometric mean?
Because multiplication by zero destroys the product however large the other values are. That is a feature rather than a bug: a single year of total loss really does wipe out everything that was compounded before it.
More Comparisons
- GPA vs Weighted GPA
The same five courses give a 3.560 GPA on the credit-weighted scale and 3.760 with AP bonus points — a 0.200 lift. One weights by credit hours, the other by how hard the course is.
- Average vs Median
The average uses every value; the median uses the middle one. On five incomes where one is an outlier, the average moves 8.65 and the median moves 1.5 — here is why that matters.
- Percentile Rank vs Z-Score
A score of 78 against a mean of 70 and a standard deviation of 8 is a z-score of 1.00 and the 84.13th percentile — 421 of 500 test takers below it. One number is measured in standard deviations, the other in people.
Sources
- NIST/SEMATECH e-Handbook — measures of scale and location — National Institute of Standards and Technology
- Geometric mean — definition and use for growth rates — Wikipedia
- Weighted arithmetic mean — Wikipedia
Both methods sit in Math.