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I'm trying to calculate the percentage difference $\Bbb{P}$ between two numbers, $x$ and $y$. I'm not given much context about how these numbers are different (for example, if one is "expected outcome" and one is "observed outcome").

When is each of this formulas relevant?

$$\Bbb{P} = \frac{|x-y|}{x}, \Bbb{P} = \frac{|x-y|}{y}, \Bbb{P} = \frac{|x-y|}{max(x,y)}, \Bbb{P} = \frac{|x-y|}{\frac{x+y}{2}}$$

What is the difference between the first two? Is the last one the most general?

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All of these can be relevant in a certain context. The 1st one is the percentage difference as a percentage of $x$, the 2nd one is the percentage difference as a percentage of $y$ the 3rd one is the percentage difference as a percentage of the max of $x, y$ and so on.

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The word 'change' implies a beginning state, and an ending state. If we have an observation $x$, that is then realized as $y$ in the future, then the percentage change of $x$ to $y$ would be $$100\%\cdot \frac{y-x}{x}$$

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