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Say I have a collection of ordinal data, such as that based on a Likert scale. The underlying data might be from a set of the following values:

  1. Strongly disagree
  2. Disagree
  3. Neither agree nor disagree
  4. Agree
  5. Strongly agree

As part of the analysis, I might present the median of this data. If the median falls between two values (e.g. between 3 and 4), what is the best way of presenting this?

I've seen an academic paper report this as "3.5", but I'm not sure if this is appropriate. My understanding would be that, because this is ordinal data, this might imply to readers that the median is roughly halfway between these values, even though that isn't necessarily the case.

I might be overthinking this - just wondering how this is typically reported. Maybe "3-4" is less misleading?

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    $\begingroup$ I would add an explanatory note somewhere. If you're not using the medians for any purpose beyond reporting them, 3-4 and 3.5 with a note both have some defence. $\endgroup$
    – Nick Cox
    Aug 22, 2021 at 11:37

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Sample quantiles such as the sample median require the distribution to be continuous. Otherwise the addition of a single observation can move the quantile by a whole unit, which is not sensible. For single variable summaries of ordinal variables you are left with these options:

  1. Temporarily assume that the measurement scale is interval (equal spacing of categories) and use the mean (that's what I would use here)
  2. Compute 5 proportions
  3. Compute 4 cumulative proportions

When you get to multivariable summaries the ordinal nature of the data can be used to its fullest, e.g., rank correlations, semiparametric ordinal regression models such as the proportional odds model.

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    $\begingroup$ I don't follow the logic in the initial claim. A median can be a perfectly fine descriptor for discrete (even ordinal) data, even though it might be sensitive to small changes. "Sensible" and "sensitive" aren't the same. Notice, too, that continuous distributions may suffer from the same problem: this occurs when they have zero density in a neighborhood of their median(s). $\endgroup$
    – whuber
    Aug 22, 2021 at 14:28
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    $\begingroup$ Sorry have to disagree. The median is insensitive to small changes (you can move the probabilities without moving the median) and also too sensitive to small changes (with a lot of ties one data point can move the median a whole position). Good point about some continuous distributions. In general quantiles and quantile regression have a lot of trouble when ties are heavy. There are many examples in the medical literature where a treatment had an effect but it didn't affect the median, when Y is discrete. $\endgroup$ Aug 22, 2021 at 15:13
  • $\begingroup$ All your points are well taken and are useful criticisms of medians, but I find it puzzling that you end up seemingly arguing the opposite of the original point: namely, that the median is insensitive rather than too sensitive! $\endgroup$
    – whuber
    Aug 22, 2021 at 15:17
  • $\begingroup$ Granted; it is paradoxical. Too insensitive to gradual effects and too sensitive to a trivial effect from one observation, from being a discontinuous "counting" functional. $\endgroup$ Aug 22, 2021 at 18:09

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