Solution: Finding Covariation Between Qualitative and Quantitative Variables

The question asks for a statistical measure to determine the covariation between one qualitative variable (categorical) and one quantitative variable (numerical).

Understanding Association Measures

Different statistical techniques are suited for analyzing relationships between variables based on their type:

• Yules 'Q' and Phi coefficient '$\phi$' are specifically designed for measuring association between two dichotomous (binary) qualitative variables. They are unsuitable for the combination of qualitative and quantitative variables presented.
• Rank correlation 'r', such as Spearman's rho, is used to assess the monotonic relationship between two quantitative variables by analyzing their ranks. It does not apply when one variable is qualitative.
• Chi-square ($\chi^2$) is fundamentally a test for independence between two qualitative variables. However, it is widely applicable when a quantitative variable is divided into categories (discretized). In this context, the Chi-square test can evaluate the association between the original qualitative variable and the newly categorized quantitative variable.

Selecting the Correct Measure

Considering the options provided, the Chi-square ($\chi^2$) test is the most appropriate choice. It allows for the analysis of association when one variable is qualitative and the quantitative variable is treated as categorical through grouping. This helps determine if the distribution of the quantitative variable's categories is significantly related to the categories of the qualitative variable.