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I am trying to analyze responses from a survey. The outcome variable is named "reading_proficiency" and is dichotomous with two values 0 and 1. The data set has 3,539 observations so the column "reading_proficiency" has 3,539 observations of either 0 or 1.

I am trying to understand whether I can use the ideas of binomial distribution here. Can the variable "reading_proficiency" be a random variable?

The definition of a random variable I am using is as follows. A random variable is a variable that takes on different values determined by chance. In other words, it is a numerical quantity that varies at random.

Are the values of "reading_proficiency" truly determined by chance? Are two observations of "reading_proficiency" truly independent of one another?

If two observations are from the same survey cluster, they might have attended the same schools, been taught by the same teachers and thus have the same "reading_proficiency".

Is the fact that many observations are from the same cluster disqualify "reading_proficiency" from being a random variable?

I was reading that each observation of "reading_proficency" should be mutually independent but this is not the case for survey data, or...?

Does it mean survey data cannot be random variables?

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    $\begingroup$ Straight from Wikipedia A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain. They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. $\endgroup$
    – user2974951
    Jan 28, 2022 at 12:20
  • $\begingroup$ If your survey sample was obtained by randomly selecting subjects from a population, then perforce its results are random according to your definition. (For better definitions, see stats.stackexchange.com/questions/50.) If it was not a random sample, then you could still try to analyze the results using random variables, but you will have to justify that somehow. $\endgroup$
    – whuber ♦
    Jan 28, 2022 at 16:14

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Welcome to Cross Validated!

As user2974951 points out, a random variable could "...conceptually represent..the subjective randomness that results from incomplete knowledge of a quantity". That is, it 'seems' random to you because you simply don't have enough information about the factors that actually influence/determine the outcome.

So that answers the first part of your question: "Are the values of "reading_proficiency" truly determined by chance?"

Basically - regardless of what truly determines "reading proficiency", you do not have the information to know everything about it, meaning it can take on many different values, with a particular distribution, and you do not know what value it will take on in a particular case because you don't have enough information. So from that aspect, it's certainly a random variable. We're getting a bit into philosophy here, but essentially, if there's some variable/process that generates different results but you do not know everything about what determines an outcome (meaning you have uncertainty , it is a random variable). If this point is unclear, please ask in comments and we can discuss more.

Let's move on to the second part of your question: "Are two observations of "reading_proficiency" truly independent of one another?"

This is an interesting question (given your clustering concerns). However, let's be clear - I don't think "independence" is a prerequisite for being a random variable. For example, you can see clustering in almost all distributions - they tend to be clustered around the mean. So whatever process is generating that data must, in some way, result in most data points being quite connected/similar and close to the mean. You don't need sample independence for it to be a random variable.

So all in all, yes, your survey data is a random variable.

Perhaps you're in some way starting to confuse "random variable" with "random sample"? It's with taking "random samples" that you need to think about whether the individual samples are independent, and that's where your clustering concerns might come in.

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  • $\begingroup$ "Lack of information" does not equate with being random! It is particularly dangerous to believe this when data were not randomly sampled. $\endgroup$
    – whuber ♦
    Jan 28, 2022 at 16:15
  • $\begingroup$ @whuber But "true randomness" is a pretty abstract/philosophical concept, no? If the observer has uncertainty (due to incomplete information), then that process acts like a random variable to that observer. Right? Additionally, I don't think random sampling have an effect on whether or not something is a random variable. Take any set of samples for anything - test scores across the nation. Those scores will form a distribution and represent a random variable, even though they're not truly independent, since the kids have things in common (SES clusters, same national education system etc.) $\endgroup$
    – Vladimir Belik
    Jan 28, 2022 at 16:26
  • $\begingroup$ Re "right?" Wrong. Not all uncertainty is random. Random sampling definitely has an effect on whether one can use random variables in a model! Look into the distinction between sample uncertainty and model uncertainty. The former is created by the physical process of sampling while the latter is introduced by fiat: namely, by making assumptions. The justification of one's analysis varies greatly based on that distinction. $\endgroup$
    – whuber ♦
    Jan 28, 2022 at 16:30
  • $\begingroup$ @whuber I agree with everything you said about the modelling, and I totally agree that sampling heavily affects how it can be used in modelling. No issue there. But I can't see how a non-random sample doesn't mean that the result isn't a random variable. For example, you sample test scores from a class but you only sample the boy's scores. That's not a random sample, but your result is certainly still a random variable - it's just different than what you think it is (since it's a random variable for boys scores, not total class scores). $\endgroup$
    – Vladimir Belik
    Jan 28, 2022 at 16:36
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    $\begingroup$ I never wrote, or even implied, that one cannot use random variables for analyzing non-random samples. I have only emphasized the need for a different kind of justification for such a model, along with the possibility that it might not be justifiable in some circumstances. $\endgroup$
    – whuber ♦
    Jan 28, 2022 at 17:18

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