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When I collect data for my work, I get a set like the following:

[ 133, 183, 185.16, 188, 143, 128, 135.5, 100.55, 117.96, 95.5 ]

These points are part of a continuous scale, but there is no known function associated with them. How does one develop a probability density (continuous) function from this set?

I know I can find higher-order calculations based on this unknown probability density function, such as the value of the Expectation operator on this set by way of a basic average; however, this average requires treating the measurements as discrete values each with the same probability of 1/n, and thus this is managed as a discrete probability distribution function, even though the values are on a continuous scale.

This is unlike most discrete data problem examples, where dice are used and there is no continuous scale (we either have 1, 2, 3, 4, etc.).

This is the oddity that is bugging me: How to determine continuousness vs discreteness. If the function that describes the data is known, then my assumption is we can act on it as a continuous data set, but if not then its all simply discrete?

It is highly unlikely that within the precision of my instrumentation I will get the same value for two measurement events; however, this is entirely possible, so is this data simply a massive sample space of all possible measures (down to some finite precision), where it is possible to have overlap (though not probable)?

The stats books and examples I've dealt with so far love to outline discrete data sets as those with a great deal of obvious event overlap, such as dice, cards, and coins, where the values and events are finite and part of a rather small sample space. For continuous data sets, examples always include a function that describes the probability density over some domain range.

This leaves me wondering about this specific situation, where the measurements are clearly discrete in the sense that they are separate points on a line, but this seems to differ from other examples of discrete events, where in contrast, their values are not continuous.

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  • $\begingroup$ These are good things to wonder about. However, many of them lead to non-questions in the sense that most data have no inherent type: your model of the data determines what type they have and you are allowed to contemplate different models using different types. That leaves me scanning through your post to identify any substantial question. Would it be the first one about estimating a density? The second one that the data must be modeled as discrete if they are not modeled as continuous? The third one about modeling measurement error? $\endgroup$
    – whuber ♦
    Commented Apr 29, 2015 at 18:37
  • $\begingroup$ I wonder if you are making a mountain out of a molehill. The sample average can be used as an estimate of the population mean parameter no matter whether the population is continuous (say, Gaussian--$\mu$) or discrete but uncountable (say, Poisson--$\lambda$) or discrete & finite (say, binomial--$\pi$). $\endgroup$
    – gung - Reinstate Monica
    Commented Apr 29, 2015 at 18:41
  • $\begingroup$ @whuber, Apologies if the question wasn't clear. The second one you mention is more along the lines of what I'm wondering. Here the data is on a continuous scale, but is discretely sampled. Without a model of what happens on the entire range of the continuous random variable, are we left managing it as a discrete data set? How (if possible) would one bridge this to manage the continuous nature of the random variable? It seems this is quite relevant, especially when inferring conclusions for a population as a whole from a sample. $\endgroup$
    – Topher
    Commented Apr 29, 2015 at 18:47
  • $\begingroup$ @gung, I definitely may be ;), but am doing so for the sake of understanding the connection between simply getting the classic "average" of a data set such as the above, and approaching this from the more theoretical standpoint (as is presented in books), such as performing an expectation operation on the probability density/distribution function for this data. I know how to get the average easily, but am not seeing how to handle this as a density or distribution function, and where that distinction lies. $\endgroup$
    – Topher
    Commented Apr 29, 2015 at 18:52
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    $\begingroup$ I'm not quite sure if I follow all of that (@whuber may be able to help you more). Generally, we take the expectation of the theoretical distribution, not sample data. In mathematical statistics classes, you show that the sample average is a consistent estimator, etc. I'm not sure you need to worry about what you are trying to do here. $\endgroup$
    – gung - Reinstate Monica
    Commented Apr 29, 2015 at 18:59

1 Answer 1

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In data analysis, we usually treat any variable that has a lot of distinct ordered values as continuous, even though all real-life measurements are discrete. Ben Kuhn's answer here gives some reasons why. This is an example of how doing good data analysis is about finding a model that is useful for the task at hand, rather than finding the model with the most realistic assumptions.

Applying statistical functionals (like the mean) to an entire infinite population is the kind of thing that you get to do only in the rarefied world of mathematical statistics. In applied data analysis, you have only a sample and you'll never know the true model, so you have to settle for estimation using the wrong model, or nonparametric estimation.

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  • $\begingroup$ But if all are treated as continuous, would that not mean there has to be a model to use for continuous calculations? Without such a model, you would have to treat the data as discrete points, and still perform actions like summing and dividing to get the average, and then using this to get variance, SEM, etc., even if you have a million of them. This would all be discrete approaches to calculations, even though the measurements are continuous in nature, and aren't say, limited to integer values such as rolls of a dice? $\endgroup$
    – Topher
    Commented May 1, 2015 at 19:22
  • $\begingroup$ "But if all are treated as continuous, would that not mean there has to be a model to use for continuous calculations?" — Yes (excepting nonparametric methods). What I'm saying is that you have to use a model, even though you know it's wrong. $\endgroup$
    – Kodiologist
    Commented May 1, 2015 at 21:42

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