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I am trying to understand the difference between discrete data and continuous data in a real world application.

My example is this:

Once a day I do pushups until my body is tired. I am counting how many pushups I do and recording it. I do not count partial pushups.

day n_pushups difference
1 15 null
2 12 -0.2
3 15 0.25
4 18 0.2

My understanding is that the n_pushups is a discrete variable since it is "measurable" and "exact."

However, I decide to enrich this data by creating a view (column) from it that shows the percentage difference from the previous day's pushups.

I am measuring a discrete variable n_pushups and then enriching that data to create a new set of variables that appear to be continuous. I have a few questions:

  1. Is my observation correct that counting is discrete and the percentage difference is continuous?
  2. Is it acceptable to begin with discrete data, enrich it to get continuous data, and then apply continuous methods of analysis on this new data?

More detail on question #2: I may want to apply some analysis on my pushups that involve the normal distribution, but I know I cannot do that for discrete data. As a proxy, I was thinking I could analyze something about that data that I could then apply the analysis to.

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Great question! And one that explores a "simplifying lie" that's often told in intro statistics classes.

Commonly, when giving examples of discrete data to contrast them with continuous data, we often say something like "If you have numbers like "1, 2, 3" they're discrete, if you have decimals like "1.337, 3.1415, 2.71" those are continuous.

But actually, continuity vs discreteness has to do with which numbers are possible.

In your case, though the percentages have decimal points, you can't get any possible percentage difference. (For example, the percent difference will never be $\frac{1}{\sqrt{2}}$). You can only get percentages expressible as some reasonable ratio of pushups. So rather than a continuum of percentages, which would be continuous data, you have discrete data in disguise!

It is not possible to apply a simple transformation to discrete data to obtain continuous data.

So can you apply continuous methods of analysis to those data, even though they're technically discrete? Yes! We do this all the time. There's a technically more precise way of handling those data that respects their discreteness, but the overwhelming majority of cases in practice I would wager it is not used.

You ask specifically about normal assumptions. The percent differences may or may not be appropriately modeled with a normal, we need to examine them to find out.

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  • $\begingroup$ Your statement, "So can you apply continuous methods of analysis to those data, even though they're technically discrete? Yes! We do this all the time." is what I was trying to understand. Thank you! $\endgroup$
    – Jesse H.
    Aug 5, 2022 at 20:22

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