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I have a set of data about some students and I want to check if their success in exams is determined by their level of technological knowledge, or their school. So, I have a table with those values (left) and I converted into numeric values (right):

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I did this so I can apply correlation and analyze if there are a strong relation between variables, like this way:

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This let me see that when the note is good, the users are generally of basic level. And that students of APS use to have good notes. But there is no sense with the middle matchings (-0,57).

The thing is: is it ok to perform such transformations to the original data? or should I use another method for knowing the level of relation between a qualitative data and quantitative one? Thank you!

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If you're using Pearson's correlation coefficient ($r$, the most widely-used metric), then this isn't the best method. Pearson's $r$ measures the strength of the linear relationship between two quantitative variables.

In changing the qualitative variable to a quantitative one, you have to be careful. For example, changing "basic," "intermediate," and "expert" to 30, 60, and 90, respectively, you are in essence saying that the difference between levels "basic" and "expert" is exactly twice as large as the difference between "basic" and "intermediate" as well as exactly twice as large as the difference between "intermediate" and "expert."

A more appropriate way to do this is to use a linear model (regression) and use indicator variables. When you interpret the coefficients in your model, you will have one level as your "reference level." You can think of this as your baseline. The coefficients of the other two levels will indicate the average difference in your outcome $Y$ between your reference level and the level corresponding to that coefficient.

If your model is $[Exam Note] = \beta_0 + \beta_{intermediate}X_1 + \beta_{expert}X_2$, then $\beta_{intermediate}$ indicates the average difference in exam note between basic and intermediate skill levels. This website may be helpful in explaining linear regression with categorical predictor variables: http://www.psychstat.missouristate.edu/multibook/mlt08m.html

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