How does the probablity a sandwich is more than 12 inches relate to how long the sandwich actually is?

I'm having a lot of trouble googling this because I'm not sure what the correct term is for this problem. If I have a logistic regression model that gives the probability a sandwich is more than 12 inches how do I use that probability to estimate how long the sandwich actually is? If the model says the sandwich has a 65.5% chance of being more than 12 inches long does that imply a length the sandwich actually is? Perhaps I use the mean and standard deviation of known sandwiches to somehow relate the 65.5% probability to actual sandwich length?

The minimum sandwich length is 7.86 inches: if your sandwich probability mass is such that the sandwich is of zero length with probability 34.5% and length 12 inches with probability 65.5%. And note that this already presupposes that there are no sandwiches of negative length, which sounds reasonable for sandwiches, but may not hold in non-sandwich applications.

On the other hand, your sandwich can be of any expected length larger than that, simply by shifting the probability mass of this simple example to the right of 12 inches. Or by assuming pretty much any other distribution.

You can't even simply assume a normal distribution, because many distributions with different means and standard deviations are consistent with having a 0.345 quantile at 12 inches, e.g., a sandwich with mean length 12.4 and standard deviation 1, or one with mean 16 and standard deviation 10.

So you will need to add some more knowledge of your problem before you can say anything.

• It sounds like what is needed is the probability mass function? I can use the mean and standard deviation of the sandwiches in the training data, assume a normal distribution, and create a probability mass function to estimate how long a sandwich with a 65.5% probability of being over 12 inches is.
– Eric
Commented Mar 22, 2021 at 19:40
• Yes, if you can fit a distribution to your sandwiches, and if you can assume that this distribution applies to all sandwiches, then the expectation you are looking for is the one of a truncated distribution (search for that term, together with the distribution you are fitting). If you fit a normal distribution, then of course the distribution of sandwich lengths greater than 12 inches is a truncated normal, and you can take the expectation right off Wikipedia. Commented Mar 23, 2021 at 6:00

I'd say it is quite hard to estimate actual lengths since you do not know about the distribution of the random variable "length of a sandwich". However, maybe some of the fellow colleagues here have a different view on that problem. :)