# One or two tailed probability?

A rare bird recently turned up in the UK. The length of the bird's beak was measured (41.9mm). I want to check that the bird is likely to derive from the North American population of the species. To do this, I calculated the mean and standard deviation beak length of 30 North American individuals of the species. To check the likelihood that the UK bird was derived from the North America population of the species, I calculated the number of standard deviations the UK bird was away from the North American mean, which is 1.33 standard deviations below the mean.

The figure below visualises the beak length of the UK bird (red line) in relation to North American individuals.

I would like to express the probability the UK bird was derived from the North American population. Should I use a left-tailed probability, therefore the probability of a value 1.33 or lower than the mean (given bird was 1.33 standard deviations below mean) or a two-tailed probability, therefore a beak length 1.33 or more standard deviations shorter or longer than the mean (given I had no prior expectation the beak would be shorter than average)?

• One possibility: Calculate a ($\beta, \alpha$)-tolerance interval based on the NA-population data. In this range, you'd expect $\beta\%$ (e.g. $95\%$) of beak lengths from this population with a confidence of $(1 - \alpha)$. If the UK-data is within these tolerance limits, it's compatible with the NA-data. Commented Jun 25, 2023 at 17:13
• If it was not North American, where else might it have come from? What is the beak distribution in that other place? Commented Jun 27, 2023 at 16:51
• There are various populations in South America also. But they all have similar bill lengths, so probably not going to be able to say anything definite. Commented Jun 27, 2023 at 17:50

This question, as posed, doesn't have an answer. First of all, to compute a probability, you need a model. E.g., perhaps you'd be willing to assume that beak length is normally distributed with mean and variance estimated using the North American data. Second, you'd need to be precise about the definition of "derived from the North American population". E.g., you might say that, if the UK bird's beak length was within $$\pm3$$ SD of the mean North American beak length, that the bird was derived from the North American population. But this choice is clearly arbitrary.