Descriptive Statistics
Geometric Mean Calculator
The geometric mean is the exponential of the average logarithm, so it fits growth factors and other values that multiply rather than add.
Worksheet
Press Calculate to see the result.
The geometric mean is the exponential of the average logarithm, so it fits growth factors and other values that multiply rather than add.
What this calculator answers
Four growth factors, 1.05, 1.10, 1.02, and 1.08, combine by multiplication across four periods. Adding them and dividing by four treats a factor of 1.10 as if it were a count that can be piled on a factor of 1.02. The geometric mean takes the log of each factor, averages those logs, and exponentiates. That is the constant factor which, applied four times, matches the product of the four actual factors. A zero or a negative factor has no real logarithm, so the page stops instead of returning a complex number.
The headline is the geometric mean. The arithmetic mean of the same positive values is shown only so you can see the gap between adding and multiplying. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Geometric mean uses this relationship.
geometric mean = exp( (ln x1 + ln x2 + ... + ln xn) / n )
- ln. Natural logarithm. Defined here only for values above zero.
- exp. Exponential, which undoes the average of the logs.
- n. How many positive values were typed.
- product. The geometric mean is also the nth root of the product of the values.
The log turns products into sums. Averaging the logs and exponentiating returns a number on the original scale, and that number is the nth root of the product. The arithmetic mean of the raw factors is a different question. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The values are 1.05, 1.1, 1.02, 1.08, and each one is greater than zero. The natural logs are 0.0488, 0.0953, 0.0198, 0.077. The mean of those logs is 0.0602.
Exponentiate that mean: exp(0.0602) = 1.0621. That is the geometric mean. It equals the 4th root of the product of the values. The arithmetic mean of the same values is 1.0625, which adds the values instead of multiplying them.
The headline is the geometric mean, 1.0621. A zero or a negative input would have been rejected before a logarithm was taken.
| Geometric mean | 1.0621 |
|---|---|
| Arithmetic mean | 1.0625 |
| Count | 4 |
A second situation
The factor 1.02 is replaced by 1.40
The values are 1.05, 1.1, 1.4, 1.08, and each one is greater than zero. The natural logs are 0.0488, 0.0953, 0.3365, 0.077. The mean of those logs is 0.1394.
Exponentiate that mean: exp(0.1394) = 1.1496. That is the geometric mean. It equals the 4th root of the product of the values. The arithmetic mean of the same values is 1.1575, which adds the values instead of multiplying them.
The headline is the geometric mean, 1.1496. A zero or a negative input would have been rejected before a logarithm was taken.
| Geometric mean | 1.1496 |
|---|---|
| Arithmetic mean | 1.1575 |
| Count | 4 |
Replacing 1.02 with 1.40 raises the geometric mean from 1.0621 to 1.1496. The larger factor changes the product, so the constant equivalent factor has to rise with it. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The third growth factor changed from 1.02 to 1.40. The headline result went from 1.0621 to 1.1496.
The nudged inputs
The values are 1.05, 1.1, 1.4, 1.08, and each one is greater than zero. The natural logs are 0.0488, 0.0953, 0.3365, 0.077. The mean of those logs is 0.1394.
Exponentiate that mean: exp(0.1394) = 1.1496. That is the geometric mean. It equals the 4th root of the product of the values. The arithmetic mean of the same values is 1.1575, which adds the values instead of multiplying them.
The headline is the geometric mean, 1.1496. A zero or a negative input would have been rejected before a logarithm was taken.
| Geometric mean | 1.1496 |
|---|---|
| Arithmetic mean | 1.1575 |
| Count | 4 |
How to read the result
A geometric mean of 1.0621 on growth factors is a typical multiplier, not a typical percent from an arithmetic average of the four percents. Subtract 1 only if you truly want the equivalent growth rate. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Every value is strictly greater than zero.
- Multiplication across the values is meaningful.
- The natural log and its exponential are the pair you want.
- The count includes every period you mean to compound.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Averaging percent changes of 5, 10, 2, and 8 and calling that a geometric growth rate.
- Entering a percent as 5 instead of a factor of 1.05.
- Including a zero start value and expecting a finite mean.
- Using the geometric mean for temperatures in Celsius, which are not ratio-scale measurements.
Where a different calculator is the honest choice
Use the harmonic mean calculator for rates over equal distances. Use the arithmetic mean calculator when you want the additive mean beside both of the others. Use the mean calculator when only the additive mean is required. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Harmonic Mean Calculator, Arithmetic Mean Calculator, Mean Calculator.
Check yourself
Growth factors are 1.05, 1.10, 1.02, and 1.08. What is the geometric mean?
The geometric mean is 1.0621.
Using the result in a sentence
Successive investment factors, population growth multipliers, and fold-changes in a lab assay are multiplicative batches. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The geometric mean collapses toward zero if any single factor is tiny. That is the mathematics of a product, and it may or may not be the business question you meant to ask. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Geometric Mean Calculator commit to?
This page uses Geometric mean: geometric mean = exp( (ln x1 + ln x2 + ... + ln xn) / n ). The log turns products into sums. Averaging the logs and exponentiating returns a number on the original scale, and that number is the nth root of the product. The arithmetic mean of the raw factors is a different question. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Positive numbers, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the geometric mean is 1.0621. A geometric mean of 1.0621 on growth factors is a typical multiplier, not a typical percent from an arithmetic average of the four percents. Subtract 1 only if you truly want the equivalent growth rate. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Averaging percent changes of 5, 10, 2, and 8 and calling that a geometric growth rate. The geometric mean collapses toward zero if any single factor is tiny. That is the mathematics of a product, and it may or may not be the business question you meant to ask. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the harmonic mean calculator for rates over equal distances. Use the arithmetic mean calculator when you want the additive mean beside both of the others. Use the mean calculator when only the additive mean is required. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The geometric mean is 1.0621. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.