categorical datadescriptive statistics
Just to clarify, when I mean summary statistics, I refer to the Mean, Median Quartile ranges, Variance, Standard Deviation.
When summarising a univariate which is categorical or qualitative, considering both Nominal and Ordinal cases, does it make sense to find its mean, median, quartile ranges, variance, and standard deviation?
If so is it different than if you were summarising a continuous variable, and how?
I was able to find a diagram from this page that clarified a lot of the original confusion.
Are ordinal data treated the same as continuous data when calculating standardized z-scores?
No, they are not: When dealing with data on different measurement scales it is important that your analysis should not use mathematical operations that are not meaningful within that measurement scale. For ordinal data, only the ranking of the values in the scale is meaningful, and so you should only use operations that are invariant to all changes in the numbering of values that preserve rank-order. This counts out any operation that uses the arithmetic operations $+$, $-$, $\times$ and $\div$.
For ordinal data, the sample mean and sample standard deviation are not invariant to all changes in the numbering of values that preserve rank-order. This means that the sample mean and sample standard deviation are meaningless for ordinal data. Consequently, the z-score is also meaningless.
(Note: In some cases researchers treat apparently ordinal data as if it were interval or ratio data, which amounts to asserting that the differences/ratios in the ordered categories are meaningful. In this case there is often some argument over whether it is justifiable to treat data on a higher measurement level.)
How do I approach nominal variables when calculating standardized z-scores?
Nominal and ordinal variables do not allow use of the arithmetic operations $+$, $-$, $\times$ and $\div$, so the z-score for these variables is meaningless. For a nominal variable the only meaningful measures are those that count frequencies/relative frequencies of the categories and use the operations $=$ and $\neq$. For ordinal variables you also have meaningful measures for cumulative frequencies/relative frequencies using the operations $<$ and $>$ (taken in the order for the ordinal variable).
Best Answer
In general, the answer is no. However, one could argue that you can take the median of ordinal data, but you will, of course, have a category as the median, not a number. The median divides the data equally: Half above, half below. Ordinal data depends only on order.
Further, in some cases, the ordinality can be made into rough interval level data. This is true when the ordinal data are grouped (e.g. questions about income are often asked this way). In this case, you can find a precise median, and you may be able to approximate the other values, especially if the lower and upper bounds are specified: You can assume some distribution (e.g. uniform) within each category. Another case of ordinal data that can be made interval is when the levels are given numeric equivalents. For example: Never (0%), sometimes (10-30%), about half the time (50%) and so on.
To (once again) quote David Cox: