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I have two datasets of absolute distances to a single point in a 2D space. I have reasons to expect that if I had the sign and magnitude of these distances, my datasets would be normally distributed with a mean of zero.

Think of the datasets to be the absolute distances from the Bull's eye in a game of darts. I have a set of these distances for two different players, and I would like to compare the performance of the two players (both players had to aim for the Bull's eye).

I think the 'folded normal distribution' applies to my dataset. Since I would like to compare the two datasets, I need to answer the following questions.

  • How can I verify that I indeed have a 'folded normal distribution'?
  • How can I compare the (means of) these two datasets in this distribution? For example, am I allowed to use ANOVA?

Thank you!

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Multiply each data point by $+1$ or $-1$, taken independently with equal probabilities. If your datasets are indeed samples from folded normal distributions, you will now have normal distributions with mean $0$, and you can use any of the tools appropriate thereto.

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I like @Robert Israel's answer above for verifying your distribution as half-normal (half-normal since you believe it came from normal with mean zero, thus simpler than general folded-normal distribution). Another approach may be to compare a normalized histogram of your data to the probability density function for a half-normal distribution, which is

$$ p(x) = \frac{2}{\sqrt{\pi}\sigma}\exp\left(-\frac{x^2}{2\sigma^2}\right) $$

You'd need to use a $\sigma$ parameter, that you could perhaps empirically find using a normal fit from the ±1 approach above, or play around with the parameter if you just want to get a feel for the distribution.

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