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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.

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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.

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