# Can a variable be normally distributed on finite interval?

I am calculating rates, which can take any value between 0 and 1. Can it be normally distributed even though the domain is not the real numbers?

Thank you very much for the answers, here I represent the means of the data which are fitted a normal distribution on. I created something like 1000 means of the data using bootstrapping.

As for the raw data, it is indeed heavily skewed with a large positive skewness value. Based on your answers, the normality for t-test can’t be assumed 100%. Instead of t-tests, I’m trying to calculate confidence intervals. I have one confidence interval for the prediction using bootstrapping, although I’m not a 100% sure this is the correct way. I’m comparing 4 predictive models to decide what gives the best results. Individual predicted rates are grouped by the age of the policy and taken their average, so the predictions are for example: for the age=4 the rate = 4.2%. I want to use another method for the CI, namely the Chebyshev’s inequality. But for this I need to fit a distribution to the data. I already tried weibull, beta, gamma but none of them seem to work.

EDIT:
The model I created predicts individual rates and I take the average of these rate to get the mean rate for a group. That mean has to be estimated correctly, also be assigned a CI to it. I figured that if I perform a t-test on every group between the model predictions and the actual values that need to be predicted (test dataset) and I get not significant p-values, then the model is good. I needed this information regarding the possible normality of the values because of the t-test.

Thank you very much for all the information you’ve give me so far! You are great!

No, it cannot. At least if you by “distributed as” implies exactly. The range of the normal distribution extends from minus to plus infinity. As a practical matter, if the variance is sufficiently small, say on the order of $$(0.1)2 (0.1)^2$$, then a variable constrained to $$(0,1)(0,1)$$ can be approximately normally distributed.