Strictly speaking the answer is no because statistics depends on having a probability model. If you can do some plots and get an approximate Idea of the population distribution and if you can develop an argument for how the sample might be say a simple random sample from such a population then something approximate might be done. Have you thought of taking a probability sample large enough that the Central Limit theorem applies?. You might refer to the book in the link below. Best wishes, David
Romer, if this would be so, then almost all studies with aminal or humans (e.g. clinical sudies) would be worthless. I don't think that this is the case.
Inferential statistics, unlike descriptive statistics, is a study to apply the conclusions that have been obtained from one experimental study to more general populations. This means inferential statistics tries to answer questions about populations and samples that have never been tested in the given experiment.
Inferential statistics infer from the sample to the population, they determine the probability of characteristics of a population based on the characteristics of your sample, they also help assess the strength of the relationship between your independent variables, and your dependent (effect) variables. With inferential statistics, you can take the data from any samples and make generalizations about a population.
There are two main areas in case of inferential statistics, estimating parameters. This means taking a statistic from your sample data and using it to say something about a population parameter and hypothesis tests, this is where you can use sample data to answer research questions.