 Growth factor (mathematics)

For positive quantities measured at successive steps, the growth factor $g_i=Q_i/Q_{i-1}$ is a dimensionless ratio. It can be below one when the quantity shrinks. Repeated multiplication givesso the common factor reproducing the endpoint after $n$ identical steps is their geometric mean. A factor $g$ corresponds to a relative change $g-1$; factors must be averaged before subtracting one when summarizing compounded change.

For example, increasing $200$ by ten percent gives $220$, then decreasing the result by ten percent gives $198$. The factors are $1.1$ and $0.9$, whose product is $0.99$. Their geometric mean is $\sqrt{0.99}\approx 0.99499$, whereas their arithmetic mean is one. Thus equal positive and negative percentage changes need not cancel when applied successively to different bases.