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I was reading up on the use of Likert scales and how to analyse data. I found this bit of text confusing:

To properly analyze Likert data, one must understand the measurement scale represented by each. Numbers assigned to Likert-type items express a "greater than" relationship; however, how much greater is not implied. Because of these conditions, Likert-type items fall into the ordinal measurement scale. Descriptive statistics recommended for ordinal measurement scale items include a mode or median for central tendency and frequencies for variability. Additional analysis procedures appropriate for ordinal scale items include the chi-square measure of association, Kendall Tau B, and Kendall Tau C.

Likert scale data, on the other hand, are analyzed at the interval measurement scale. Likert scale items are created by calculating a composite score (sum or mean) from four or more type Likert-type items; therefore, the composite score for Likert scales should be analyzed at the interval measurement scale. Descriptive statistics recommended for interval scale items include the mean for central tendency and standard deviations for variability. Additional data analysis procedures appropriate for interval scale items would include the Pearson's r, t-test, ANOVA, and regression procedures

Why is it that when we take a mean of several ordinal items we can suddenly treat it as interval data?

Jamy
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    To be able to *add* the items (the average is a sum divided by a the number of terms), you must already have interval scales on the items. When you set the result of "1" + "4" + "5" to be the same outcome as "3" + "3" + "4" and you set "2" + "5" to be the same as "3" + "4" and so forth, you put constraints on the gaps between the outcomes. When you put all those equality assertions together, you're left with the following having to be true "5" - "4" = "4" - "3" = "3" - "2" = "2" - "1". That is, if you want to add the items in the first place, they must already be on an interval scale. ... ctd – Glen_b Aug 17 '21 at 13:31
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    ctd ... Without that you have no basis to claim that "1" + "4" + "5" is the same as "3" + "3" + "4" etc. – Glen_b Aug 17 '21 at 13:31
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    Indeed. You must not take means of individual items, because they are ordinal scale, but you can take means across several items, because that is white magic which transforms them into interval scale. The argument is nearer faith than logic. – Nick Cox Aug 17 '21 at 13:32
  • I suppose the underlying theme is that if you have enough detailed information, you can make fine distinctions. I think the whole business of what you cannot do/must not do with particular scales is largely myth and misinformation. Pragmatically, means of ordinal scales can be helpful, but watch out. More at https://stats.stackexchange.com/questions/67551/calculate-mean-of-ordinal-variable – Nick Cox Aug 17 '21 at 13:36
  • Note that Spearman correlation, which everyone agrees to be defensible for ordinal data, is based on putting ranks into Pearson correlation and working with their means and SDs! I.e. it's equivalent to treating ranks as interval scale. – Nick Cox Aug 17 '21 at 13:39
  • To me the bottom line is that even if summing ordinal values is OK, the resulting scale is likely to have a strange distribution, e.g., floor or ceiling effect, bimodality, etc. So it needs to be analyzed as ordinal (e.g., with a semiparametric model). – Frank Harrell Aug 19 '21 at 12:25

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