# What is the distribution of a data set

I understand what the probability distribution is.

I also have a personal understanding/interpretation of the concept of distribution of a dataset. Whenever I see this expression I imagine a graph with frequency as the y-axis and the members of the data set on the x-axis, for each of them(members of the data set) the graph containing a point at the corresponding frequency level.

1. Is this the correct interpretation ? Is "distribution of a datset" = "probability distribution" ? To me it doesn't look like the two concepts are the same thing.(probably subtly related but not the same thing)

2. I was unable to find a standard definition of this concept. Can you provide me with a pointer to a resource defining it ?

3. When authors say: "Two data sets drawn from the same underlying distribution", what exactly do they mean by "underlying distribution" ? Do they mean the same thing as I mentioned above, i.e. a graph like :frequency vs each member of the data set ?

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I'm still waiting for answers. In case there are unclarities in the way the question is phrased, please let me know. – Razvan Nov 16 '12 at 17:25

You understand the concept of a probability distribution, so let's start there.

A probability distribution has a cumulative distribution function that gives us the probability that a variable is less than or equal to a given value. In the discrete case, this CDF is the sum of values at discrete points of the probability mass function; in the continuous case, it is the integral over the real line of a probability distribution function.

In either case, the pmf/pdf is non-zero and consequently its sum/integral is monotonically non-decreasing to 1.

From the pmf/pdf, we can obtain distribution moments in the typical way: expected value, variance, and higher-order moments, using the standard formulas which need not repeat here. One way of looking at this is that you can characterize a distribution in terms of its moments. A Gaussian distribution is parameterized by its mean and variance; a Poisson distribution is parameterized by its process intensity. You still need to know the shape of the distribution function, but if you do know that, all you need is a handful of parameters.

(Actually, there are other ways that we can address this when you don't know the distribution!)

Now, let's look at a data set. In the real world, we really don't like dealing with a continuum. If you measure the voltage on a widget, it would be a lot easier if 5.000001 volts was effectively the same as 5.000002. Even when the physics underlying our data set dictate that the output belongs to a continuum, we want to discretize it some way.

Typically, we do this using a histogram. There are plenty of resources on how to intelligently set the bin size for a histogram, but ultimately there is no perfect, natural, context-free way to do so. As you know, a histogram counts events at discrete points.

In this way, a histogram is very much like a probability mass function: we cannot have a negative number of events in a bin, and if you add the total events from left to right, you end up counting the total number of events. Although the histogram won't sum to unity, if you just divide every count by the total number of events, you end up exactly with something that looks like a pmf.

Furthermore, you can compute statistics on the data. Mean, variance, kurtosis, etc. are all statistical moments that can be computed in a straightforward manner. In fact, there are many different types of moments that you can compute, but if you compare the formula for doing so to the canonical way of computing different, say, expected values on a pmf, they are very similar (if not identical)!

So you can take your data and turn it into something that looks like a pmf. You can even perform the same steps on the data to get statistical moments. The only thing that's really difficult to do is to find the shape of the distribution. Is it Gaussian? Binomial? Poisson? Weibull?

There are tests for showing how well your data fits any given theoretical distribution, but unless you have infinity samples, you can never say for 100% sure. Furthermore, your moment computations aren't exactly the same. Your theoretical distribution might demand discrete values at exactly 1 volt, 2 volts, 3 volts, etc., but you compute your actual sample mean using measured values; .956 volts, 2.14 volts, 2.98 volts, etc.

So in the end, a distribution of data is the characterization of the statistical moments of the data along with a comparison of the data to a theoretical distribution. Saying that data has mean and variance of X and Y doesn't give you the full picture. But saying that the data has mean X and variance Y and passes a goodness of fit test for a normal distribution does mean something, because we have tools for organizing the data in a way that has a natural analog to purely theoretical probability definitions.

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I think what you mean distribution of a dataset actually refers to the distribution of data instances. For a general dataset, you do not know the length, so if you want to define a probability density (assume continuous case) for all possible datasets, you would have to assume infinity dimension. But in practical, every instance in the dataset have a fixed length representation, which corresponds to $d$-dimensional vector, and it is easy to assign a probability density to every such $d$-dimensional point.