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I am encountering difficulties in determining the normality of data and distinguishing whether the data I have collected is parametric or non-parametric.

I have gathered data using a 7-point Likert scale, with participants providing scores for four different conditions (A, B, C, D).

If I wish to discern whether this data follows a normal distribution using the Shapiro-Wilk test, should I input the data collected for condition A, B, C, and D separately into the Shapiro-Wilk test? Or should I consider conditions A, B, C, and D as a single condition and examine the normality of all data at once?

Furthermore, if I were to enter each condition separately and find that conditions A and B conform to normality, while conditions C and D do not, should I proceed with a parametric ANOVA to compare these four conditions? Or should I employ a different non-parametric method?

I would greatly appreciate your guidance on this matter.

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    $\begingroup$ In your particular case, the Likert scale means that the data can't be normally distributed, as the answer from Dave points out. In general, you might want to look at this page on normality testing. It can be useful for evaluating residuals between observations and regression model predictions, but even with continuous outcome variables it's not useful for evaluating the variable values themselves. For your type of problem, consider ordinal regression. $\endgroup$
    – EdM
    Commented Nov 6, 2023 at 16:21
  • $\begingroup$ Data are neither parametric nor nonparametric. Models are. The term parametric relates to the number of parameters in the model (if it's fixed and finite, the model is described as parametric). E.g. If your model for a set of data was an exponential distribution - with its one parameter, your model would be called parametric. Note that the definition of parametric contains no reference to normality at all. $\endgroup$
    – Glen_b
    Commented Nov 7, 2023 at 4:08

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You know with certainty that your distribution is not normal. A normal distribution can take any real number. Your distribution takes one of seven values.

No hypothesis testing is necessary or even useful to address if your distribution is normal, as you know it is not.

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  • $\begingroup$ But by this logic all variables are non-normal, because no variable can be represented with more than a finite number of values. $\endgroup$
    – whuber
    Commented Nov 6, 2023 at 18:00
  • $\begingroup$ @whuber I'm not sure I see an issue with that. $P(\mathbb R \setminus\mathbb Q) = 0$ doesn't exactly scream out normality. $\endgroup$
    – Dave
    Commented Nov 6, 2023 at 19:45
  • $\begingroup$ IMHO your argument misses the main statistical issues, which include (1) the distribution of the variables doesn't matter -- only the sampling distribution of the statistics does -- and (2) Normality in a pure mathematical sense is irrelevant, too, because all that matters is whether the sampling distribution is sufficiently close to Normal to justify a Normal approximation for computing CIs, p-values, etc. Since we can with great effectiveness use a Normal approximation even for binary data, perforce it potentially can work even better for Likert data. $\endgroup$
    – whuber
    Commented Nov 6, 2023 at 20:25

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