Use the geometric mean for growth rates when each period changes the amount left by the previous period. It finds one constant rate that reaches the same final amount. Take the geometric mean of the growth factors, such as 1.20 and 0.90, then subtract 1. Don’t average the signed percentages themselves.
A percentage change acts on a starting amount. When that starting amount changes between periods, adding the percentages misses part of the story. A rise of 20% followed by a fall of 10% turns 100 into 108. The arithmetic average is 5%, but two rounds of 5% growth would turn 100 into 110.25.
This lesson explains why that gap appears, how to calculate the compound average, and which conditions make the result useful. The examples use a simple measured amount, with no additions or removals between readings. For related lessons and tools, visit the statistics and probability hub.
Why Do Growth Rates Multiply?
Growth rates multiply because each percentage changes the current amount. A 20% rise multiplies it by 1.20. A 10% fall multiplies the new amount by 0.90. The combined factor is therefore 1.20 × 0.90 = 1.08. The whole sequence produces an 8% increase, regardless of the original positive amount.
Think of the factor as the share of the amount that remains after a change. A factor above 1 means growth. A factor below 1 means a decline. A factor of 1 means no change. This way of writing a rate keeps both the old amount and its change together.
For example, 20% growth includes all of the old amount plus another fifth. That gives 100% + 20% = 120%, or 1.20 times the old amount. A 10% decline leaves 90%, so its factor is 0.90. Use decimals in the calculation, and convert the answer back to a percentage at the end.
The University of Toronto’s explanation of the geometric mean distinguishes an average that preserves a sum from one that preserves a product. That distinction answers the growth question: we need equal factors that preserve the final amount, rather than equal percentages that preserve their sum.
How Do You Calculate Geometric Mean Growth?
First change each percentage rate into a factor. Multiply the factors, take the root for the number of equal periods, and subtract 1. The root gives the average factor. Subtracting 1 gives the average compound rate. These are different quantities, so label the result clearly.
Average factor: G = ((1 + r1)(1 + r2)⋯(1 + rn))1/n
Average compound rate: g = G − 1
Here, r is a rate written as a decimal, and n is the number of equal-duration periods. The ordinary method uses positive factors. For example, enter a 10% decline as r = −0.10, which gives a positive factor of 0.90. Don’t enter the rate as −10.
NIST defines the geometric mean as the nth root of a product of n values. In this growth calculation, those values are the factors. Applying that definition to raw signed percentages would answer a different question and may not produce a usable real result.
The reason for taking a root is simple: the same factor must work n times. If G is the constant factor, then Gn must equal the actual combined factor. The nth root solves that equation. This also gives a quick check: raise your average factor to n and compare it with the original product.
What Happens With A Gain And A Loss?
A gain of 20% followed by a loss of 10% has an average compound rate of about 3.923% per period. Starting from 100, the first change reaches 120 and the second reaches 108. A constant factor of √1.08 reaches that same endpoint after two periods.
| Path | Start | After period 1 | After period 2 |
|---|---|---|---|
| Actual changes: +20%, then −10% | 100 | 120 | 108 |
| Constant compound rate: about +3.923% | 100 | About 103.923 | 108 |
| Arithmetic average: +5% each period | 100 | 105 | 110.25 |
The calculation is g = (1.20 × 0.90)1/2 − 1 = √1.08 − 1 ≈ 0.0392305. Multiply by 100 to express the rate as about 3.923%. The total growth is 8%; the compound average spreads that growth across two equal periods.
The equal-rate path is a summary, not a record of what happened. It replaces a rise and fall with steady growth. Keep the original rates when the timing or size of changes matters. Two sequences can have the same compound average while showing very different changes along the way.
A shorter contrast makes the same point: +25% followed by −20% returns 100 to 100. Its compound average is 0%, even though its arithmetic average is 2.5%. The two percentages act on different bases. A percent gain and a percent loss don’t cancel simply because their signs differ.
How Does It Work Across Three Periods?
For three equal periods, multiply three factors and take the cube root. Consider +10%, 0%, and −10%. Their factors are 1.10, 1.00, and 0.90. The product is 0.99, so the average compound rate is 0.991/3 − 1, or about −0.33445% per period.
Starting with 100, the amount moves to 110, stays at 110, and then drops to 99. The arithmetic average of the three rates is 0%. But repeating a 0% rate would leave the amount at 100. It would miss the small loss shown by the actual changes.
Keep the full factor until the last step to reduce rounding error. Rounding 0.996655 to 1.00 too early would erase the decline. A result close to zero still matters when you check an endpoint. Always say whether your displayed rate is exact or rounded.
How Do Period Counts And Time Units Work?
Count elapsed periods, not the number of recorded amounts. Readings at the start, after one year, and after two years describe two annual periods. If you know the positive start and end amounts, the compound rate is (end ÷ start)1/n − 1 for n equal periods.
This endpoint formula works because adjacent ratios cancel. Multiplying amount 1 ÷ amount 0 by amount 2 ÷ amount 1 leaves amount 2 ÷ amount 0. With annual periods, the result is a compound annual growth rate, often called CAGR. Monthly periods instead produce a monthly compound rate.
Unequal periods need a shared time unit. Multiply their growth factors to find the full change, then take a root based on total duration in that unit. Don’t treat a three-year rate and a one-year rate as two equal annual observations. Their labels must guide the calculation.
For example, a total rise of 10% over three years has an equivalent annual rate of 1.101/3 − 1 ≈ 3.228%. It is not 10% per year. Likewise, if one interval lasts two years and another lasts one, their combined factor needs a cube root to give a yearly average.
You can check an interval’s factor with the percentage change calculator. To explore repeated growth at one fixed rate, use the compound interest calculator. The geometric mean handles the separate task of finding an equivalent constant rate from a sequence of changes.
When Does The Method Need Care?
The usual growth interpretation assumes a positive starting amount and positive factors. A negative rate can fit this model: −10% leaves a factor of 0.90. A factor below zero does not describe ordinary growth of a positive amount. Don’t force such a sequence into this method.
A zero factor is a boundary case. It means a total loss, leaving a product of zero and an endpoint of zero. The root formula then gives a constant factor of zero, or a −100% equivalent rate. However, a logarithm-based calculation cannot use it because the logarithm of zero is undefined.
The zero case also shows the summary’s limits. Once the amount is zero, a later percentage increase cannot restore it without something being added. The original timing still matters. An equivalent factor of zero does not explain which period caused the total loss.
Outside additions or removals require care as well. If an amount rises because someone adds more to it, the endpoint ratio includes that addition. A withdrawal can lower the endpoint without showing weak growth. Before interpreting an endpoint rate as performance, separate outside flows from changes in the amount itself.
Choosing The Right Average
Choose the geometric mean when your question is, “What constant rate would reproduce this final amount?” Choose the arithmetic mean when your question concerns the simple average of the listed period rates. Both can be valid descriptions, but they preserve different things and should not share an unclear label.
The arithmetic mean adds rates and divides by their count. It can describe the average observed rate for a single period in your data. It does not, by itself, reproduce a chain of changes. The mean, median, and mode calculator supports that separate question about the data’s ordinary average.
Before reporting a compound average, check three things: each rate belongs to a clear period, each rate becomes the correct factor, and the repeated average factor reaches the same endpoint. State the time unit and rounding. Keep total growth separate from growth per period so readers can compare like with like.
Finally, a geometric mean describes the sequence you supplied. It does not predict the next rate or promise future growth. Use it to summarize how multiplication changed an amount over a known span. Keep the original period data nearby whenever the path matters as much as the final value.