A college reports annual changes of plus twenty-five percent and minus twenty percent. Their arithmetic average is plus two and a half percent. Yet an enrolment count following that path returns exactly to its starting value. The apparent contradiction disappears when growth is treated as multiplication of successive factors rather than addition of percentages.

The Bureau of Economic Analysis explains compounding when converting quarterly changes into annualised rates. The calculation below independently applies the same multiplicative principle across annual observations. It uses fictional enrolment counts and does not reproduce BEA economic data or imply that an enrolment path forecasts a student's opportunities.

Follow the count through each change

Assume a fictional programme begins with 1,000 students, rises to 1,250 after one year and returns to 1,000 after the second. The first change is 250 divided by 1,000, or twenty-five percent. The second is minus 250 divided by 1,250, or minus twenty percent. Their bases differ.

Observation Enrolment Change from previous year
Start 1,000 Not applicable
Year 1 1,250 +25%
Year 2 1,000 −20%

The arithmetic average is (25 − 20) divided by two, or 2.5%. Applying that rate for both years would produce 1,000 times 1.025 squared, or 1,050.625, which does not reproduce the observed endpoint. An average of annual percentage changes is therefore not generally the constant compounded rate connecting the starting and ending counts.

To find that constant rate, solve 1,000 times (1 + g) squared equal to 1,000. The two-year compound annual growth rate is zero. More generally, for positive starting and ending counts separated by n annual intervals, g equals the nth root of ending count divided by starting count, minus one.

Count intervals, not printed years

Three observations in the table contain two annual intervals. Using three as the exponent denominator would confuse the number of data points with the length of the growth period. For a real table, record the exact start and end reference dates before calculating an annualised measure.

A second fictional path, 1,000 to 1,000 to 1,000, has the same zero compounded rate but none of the first path's variation. The endpoint rate alone cannot show the temporary expansion and reversal. Keep the intervening counts when the question concerns capacity, course demand or recent conditions.

Do not extend the formula casually to a zero starting value. Division by zero makes this expression undefined, and a move from no recorded students to a positive count requires a different description. Likewise, an apparent negative enrolment is a data or interpretation problem, not a meaningful input to an ordinary enrolment growth calculation.

Check whether the series still measures the same programme

Compounding cannot repair changes in programme boundaries, reporting coverage or counting units. A merger or a recode can change the level without representing organic growth in a comparable population. Identify breaks and preserve the source notes before attaching a precise annual rate.

Also separate historical description from prediction. A zero endpoint rate does not imply that next year's intake will be unchanged. A positive rate does not establish that teaching capacity or entry places grew at the same rate. Those questions require their own evidence.

The finished comparison should report the endpoints, interval count, compounded rate and observed path. In this example, enrolment ends where it started even though the arithmetic average annual percentage change is positive. Showing both measures with their meanings prevents an attractive average from describing growth that the actual counts did not achieve.