This problem is encountered by me lot of times.
Let's say I have a set of data points (x1, x2,....,xn) whose mean and standard deviation can be calculated. However these data points are non-normally distributed. Now, given a new data point (X_new), how do I infer how likely it is to observe this new data point being drawn from the same population as the above obtained data points.
Per my understanding, if the underlying distribution is normal in that case we can use the mean and standard deviation obtained from the observed sample to make inference about the new data point's likelihood of belonging to same distribution. However, I am not sure how do we proceed if we are not sure about the underlying distribution being normal or if they are non-normal?
Use case: I am working on a Business use case where I have historical performance of Items being sold on my website (univariate distribution of #units sold). Now given the sales of any item, I want to establish if their sale is significantly higher compared to most of the other items. The histogram of the sales is right skewed with 5% of items contributing to more than 80% sale and hence the distribution is non normal. Due to non-normality of the distribution I can not use the mean sales and the standard deviation to estimate the likelihood of any single item's sale being as high as observed and am not sure what should be my approach here.