Endpoint growth rates and percent change
An initial and final value can describe several different kinds of change. Absolute change retains the original unit, total percentage change scales that difference to the initial value, and compound growth gives the constant per-period rate that links the endpoints.
The linearized result divides total percentage change by the number of periods. It is not an arithmetic average of observed period-by-period rates because no intermediate observations are supplied.
How to calculate growth between two values
- Enter endpoints: Use comparable values with the same unit and measurement definition.
- Count intervals: Enter the number of equal-length periods between the two observations, not simply the number of labels.
- Calculate: Compare absolute, total percentage, linearized, and compound results.
- Choose the appropriate rate: Use compound rate only when a constant multiplicative endpoint equivalent is meaningful.
Formula and variables
The compound rate is the constant geometric rate per period that would transform the initial value into the final value over n equal periods.
Total growth = (Vf/Vi − 1) × 100%; compound rate = [(Vf/Vi)^(1/n) − 1] × 100%- Vi — Initial value
- Positive starting observation (quantity)
- Vf — Final value
- Nonnegative ending observation (same as Vi)
- n — Periods
- Number of equal-length intervals between observations (periods)
Growth from 100 to 150
A quantity rises from 100 to 150 over 12 equal periods.
- Initial and final
- 100 and 150
- Periods
- 12
- Total growth = (150/100 − 1) × 100 = 50%
- Compound rate = [(1.5)^(1/12) − 1] × 100
Result: Absolute growth is 50 and compound growth is approximately 3.44% per period.
A constant 3.44% multiplicative increase for 12 periods links the two endpoints.
Understanding your results
Endpoint rates hide the path
Many different time series can share the same initial value, final value, and compound endpoint rate.
- Positive results indicate growth; negative results indicate decline.
- The period unit must be stated, such as month, year, or generation.
- Volatility, seasonality, and intermediate reversals are not represented.
- A final value of zero produces a compound decline of −100% per period only in this endpoint-equivalent model.
Assumptions
- Initial and final values are comparable and use the same definition and unit.
- The initial value is positive and final value is nonnegative.
- Periods have equal length when interpreting the compound rate.
Limitations
- Does not calculate period-by-period arithmetic mean, regression trend, uncertainty, or confidence intervals.
- Does not account for additions, withdrawals, migration, inflation, seasonality, or changing measurement definitions.
- Negative endpoint values are outside this geometric model.
Common mistakes
- Counting observations instead of intervals between them.
- Calling the linearized endpoint change an average of actual period rates.
- Comparing endpoints with different units or definitions.
- Assuming the compound result describes the observed path.
Practical use cases
Scientific endpoint comparison
Normalize positive population, concentration, or measurement changes across periods.
Index and activity comparison
Express change relative to a positive starting level.
Frequently asked questions
Is compound growth the same as average growth?
It is the constant geometric endpoint rate. It is not the arithmetic average of observed rates unless additional conditions happen to make them equal.
Why must the initial value be positive?
Percentage and geometric growth divide by the initial value and are not defined by this model for zero or negative starts.
What is one period?
It is whichever equal interval your data use—such as one month, year, or generation—and must be stated with the result.
Sources and review
- Calculating Percent Changes — U.S. Bureau of Labor Statistics. Accessed 2026-07-13.
- How Is Average Annual Growth Calculated? — U.S. Bureau of Economic Analysis. Accessed 2026-07-13.
Reviewed 2026-07-13.