= Growth factor
{disambiguate=mathematics}
{title2=$g_i=Q_i/Q_{i-1}$}
= Growth factor
{synonym}
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> gives
$$
Q_n=Q_0\prod_{i=1}^n g_i,
$$
so 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 $\sqrt{0.99}\approx0.99499$, whereas their <arithmetic mean> is one. Thus equal positive and negative <percentage> changes need not cancel when applied successively to different bases.
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