Percentage within each other?

Question

Consider finding what percentage the values $15$ and $23$ are within each other. How would I calculate this? To me, it is not clear whether to do $\frac{23-15}{15}$ or to do $\frac{23-15}{23}$ to find this answer. Clearly, both calculations give different results. Hence, how does one find what percentage the two values are within each other? Does it require calculating the mean of $23$ and $25$? A clear explanation would be great.

Answer 1

The cold hard fact is that "$x$ is within $p\%$ of $y$" is not a symmetric statement in $x$ and $y$. If you say "They are within p% of each other", then we can make things precise by interpreting it to mean that $y\in[x(1-p/100),x(1+p/100)]$ and $x\in[y(1-p/100),y(1+p/100)]$.

Under this interpretation, you may be making a true statement, but it will generally not be the strongest statement possible, since as you have noticed, as long as $x$ and $y$ are unequal (and for simplicity, nonnegative), $|\frac{y-x}{x}|$ will not equal $|\frac{x-y}{y}|$. The strongest statement possible using "within" is that $x$ is within $|100\frac{x-y}{y}|\%$ of $y$, and $y$ is within $|100\frac{y-x}{x}|\%$ of $x$.

Applying this to your example, you could say that $23$ is within (approximately) $53\%$ of $15$, and $15$ is within $35\%$ of $23$. It would also be correct (though not the strongest statement possible) to say they were within $35\%$ of each other, though I don't know that people usually speak in those terms, largely because of the issue you've discovered.

So to sum it all up, if you want a single answer, and interpret "within $p\%$ of each other" as above, you'll want to take the minimum of the two percentages calculated. But you lose some information this way.