# How to determine the sample distribution based on a survey involving six variables?

This question might be too naive, but I need to understand this point. Suppose I ran a survey for a product for 1000 individuals & collected the data for various aspects of it. Let's say the categories are X1, X2, X3, X4, X5 and X6.

So, now I have 6 variables. I want to know the type of the distribution of the sample.

My question is how to know the distribution of this sample.

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There is no single answer for your question, but you can approximate the six distributions to a varying degree of accuracy. First thing you should do is plot them using either histogram (hist() in R) or a kernel density estimate (density()). It should give you and idea as to what parametric family (exponential, normal, log-normal...) might provide you with a reasonable fit. If there is one, you can proceed with estimating the parameters.

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Thanks Alex! I just want to clarify myself a bit more. According to what you are saying, I've to plot histogram (or any other approach) for each variable & then decide the parametric family of the sample. And presume other variables have the same parametric family.Or I've to plot histogram for each variable individually to get the decide the parametric family for each variable? –  Ari Aug 9 '11 at 11:58
You should do that for each variable, as you do not know for sure that they're equally distributed –  nico Aug 9 '11 at 13:39
Thanks Nico! Things are now much clearer to me. –  Ari Aug 9 '11 at 13:57
@Alex:I'm marking this as the right answer even though StasK also gave a part of the answer, as your answer is closest to what i was looking for.But I wish I can mark both the answers! –  Ari Aug 9 '11 at 18:03

I personally think this is a poor idea. If you know that your data comes from a certain distribution, you can probably say something meaningful. You may have 0/1 responses, so the distribution is binomial, may be conditional on some other covariates -- that's a logistic regression. You may have counts, so the distribution is Poisson, may be conditional on some other covariates -- that's Poisson or zero inflated Poisson or negative binomial regression. However, generally just peeking and the data and trying to determine the distribution rarely leads to good results.

Telling us what your ultimate goal of analysis is may help suggesting some better routes. Do you want to simulate new data from a similar distribution? Do you want to provide an analytical summary that's easy to compute for certain distributions? (I've seen people fit lognormal curve to income data, so as to report the Gini coefficient.) Do you want to compare your results with somebody else's?

Also, keep in mind that a small sample (say under 100) will be compatible with many possible distributions. A distribution with positive values only could be represented by a gamma, or a lognormal, or a beta, or by Pearson family, and there is simply no way distinguishing between them on the basis of the data only. On the other hand, large samples (say more than 10000) won't be compatible with anything, since the real life is richer than the assumptions we make about it.

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Thanks Stask! But I still have a question. Suppose my survey have options for yes/no (1/0) & for ranking 1 to 7 likert scale. Then if I want to do logistic regression, can I run it assuming that I've binomial distribution? –  Ari Aug 9 '11 at 13:52
Logistic regression is a binomial distribution conditional on the covariates: the log odds ratio for the probability of 1 is a linear function of the covariates. It makes all the assumptions for you, you don't have to assume anything on top of what it does already :) The model for the Likert scale is called an ordinal logistic regression. If the 0/1 part and the Likert part are determined simultaneously and are dependent on a latent variable (somebody's psychological traits, for instance), you'd have to work with some sort of an item-response model (IRT). –  StasK Aug 9 '11 at 14:51
Thanks StasK! I had this problem with basic understanding for quite sometime. Both you & Alex help me to clear this issue. –  Ari Aug 9 '11 at 15:04