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As it's commonly used, with each point representing a number, it's not a continuous variable.

For example, if someone hits a radio button for disagree=2, then it's a discrete variable.

If, however, interval choices between points are allowed by the setting, then the scale is measured and the numbers are assigned as fractions or decimals such as 1.88, it becomes a continuous variable, although still ordinal in nature as one can not infer a set ratio between each response.

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Q: Can likert scale be a continuous variable?
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