# Reconciling cumulative (total) percentage with overall percentage increase

Let's say I have the following quarterly sales: $$\begin{split} Q_1&: \100,\\ Q_2&: \120,\\ Q_3&: \140,\\ Q_4&: \110.\\ \end{split}$$ $$Q_2$$ sales are $$20\%$$ higher than $$Q_1$$. $$Q_3$$ sales are $$16.6\%$$ higher than $$Q_2$$. $$Q_4$$ sales are $$21.4\%$$ lower than $$Q_3$$ sales. The cumulative (aka total) percent change is $$20+16.6-21.4 = 15.2\%$$. However, $$\100+15.2\% = \115.2$$, which is NOT the final value (given as $$\110$$).

On the other hand, the overall change percent change can be calculated as $$(110-100)/100 = 10\%$$, which is not the same as the overall percentage change of $$15.2\%$$.

Why is that so? The reason I am asking is because I want to show that we increased sales by $$10\%$$ from $$Q_1$$ to $$Q_4$$, but then I want to break down this $$10\%$$ by quarter. But if I calculate things by quarter, I get wrong results as you saw above! How do I reconcile this discrepancy between total and overall percentage change?

Thanks.

With the 'overall percent change', you are comparing all changes to a baseline value of $$100$$. However, with the 'cumulative (aka total) percent change', you are comparing each change to the immediately previous baseline ($$100$$, $$120$$, $$140$$ respectively). But then, other than the first percent change, this means they are not the same as the changes compared to $$100$$.
Since the overall percent change is compared to the original amount ($$100$$), one could look at each change and compare each change to $$100$$. That is
$$100$$ to $$120$$ is a gain of $$20$$, which is $$20\%$$ of $$100$$.
$$120$$ to $$140$$ is a gain of $$20$$, whick is $$20\%$$ of $$100$$.
$$140$$ to $$110$$ is a loss of $$30$$, which is $$-30\%$$ of $$100$$.
And these percent changes now add up to $$10\%$$, the correct overall percent change, because we are now using the common baseline.