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I have been reading about appropriate measures of central tendency for ordinal level data. So far I have learned that the median and mode can be used but that the latter can only be used in some cases. Some sources state that the median can only be used with Likert questions when there is an odd number of scores. It is not clear to me what this means and also which cases the median cannot be used.

Example:

An example may illustrate.

  • If there was a question: "Climate change is England’s most serious environmental problem" on a response scale: 1=strongly agree 2=agree 3=unsure 4=disagree 5=strongly disagree. Would the median be 3=unsure?
  • What if no respondents stated disagree or strongly disagree and all 100 respondents stated either 1, 2, or 3, is the median then 2?
  • what if respondents only stated 2 or 3. In this case is it not possible to identify the median?
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  • $\begingroup$ This is kind of a cool question, if only because I'm sure lots of people do this without thinking about it very hard. $\endgroup$ – naught101 Aug 7 '12 at 2:27
  • $\begingroup$ Also, it's a good argument for restating questions along the lines of "on a scale of 0-10, where 10 is absolute agreement, and 0 is absolute disagreement, what do you think of the statement: ..." $\endgroup$ – naught101 Aug 7 '12 at 2:29
  • $\begingroup$ This is a duplicate of stats.stackexchange.com/questions/158920/… (and the accepted answer there is pretty good). $\endgroup$ – Stephen McAteer Sep 29 '20 at 3:08
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Definitional issues:

  • The median is the middle value of the data; it is not by definition the middle value of the scale.

  • When the sample size is even, then the median is the mean of the values either side of middle most point after rank ordering all values (see wikipedia description).

When to use median on ordinal data

  • In theory the median can be used on data from any variable where the values can be ordered.
  • In practice, the median is often not the most useful summary of central tendency with ordinal variables. This partially depends on what you want to get out of your measure of central tendency. When you are describing the central tendency of data on an ordinal variable with only a small number of response options (i.e., perhaps less than 20 or 50 or 100), the median can be quite gross (e.g., 1,1,3,3,3 and 1,3,3,5,5 both have a median of 3, but the second example would have a higher mean). When it comes to summarising the central tendency of Likert items, I find the mean to be much more useful and sensitive to meaningful differences. Ordinal variables that are ranks do not suffer from this problem of "grossness".
  • Interpolated medians are another way of overcoming the gross nature of the median on ordinal data with few values.
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    $\begingroup$ @ Jeromy. I understood that if you are using ordinal data "When the sample size is even" and the numbers on either side of the cut point are different then it is not possible to calculate the median. This is because there is no clear mid point between two ordinal values. That is, your values are ordinal not ratio/interval because there are no even distances between them. $\endgroup$ – Anne May 26 '11 at 4:22
  • $\begingroup$ Surely the mode is relevant here. And potentially more relevant than the mean, since you can't necessarily state that "unsure" is between "agree and disagree" - it could mean "I don't understand the question, but if I did, I'd bee at one extreme", or it could mean "I disagree with the premise of the question" (if there is no opt-out option). $\endgroup$ – naught101 Aug 7 '12 at 2:26
  • $\begingroup$ @naught101 In typical survey/psychological research domains, you may have a set of items all using the same scale and you want to get a sense of which items have higher or lower levels (e.g., of agreement). In that case, the mode might be interesting, but would inadequately discriminate between scores on the different items (e.g., you might have a big bunch of items all with the same mode of 4). I agree that how the ordinal scale is assigned numbers is relevant, and there are ways of empirically assessing what is an appropriate scaling. $\endgroup$ – Jeromy Anglim Aug 7 '12 at 5:10
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    $\begingroup$ That said, I think that in many applications, scoring 1,2,3,4,5 for a five-point response scale provides a reasonable first pass. In fact, in many cases, I think the exact form of scaling is not likely to make much of a difference (as long as it is a reasonable monotonic transformation) in how the items are rank ordered in terms of their means. $\endgroup$ – Jeromy Anglim Aug 7 '12 at 5:11
  • $\begingroup$ The numbers here are confusing the issue. Say our options are good (G), neutral (N) and bad (B), and the responsses are GGGGNBBB, then the median is the mean of G and N ... what is that? How can that be meaningful? I think a sensible response here is to say that the median is G/N. Maybe that's even worse? $\endgroup$ – Stephen McAteer Mar 1 '18 at 4:55
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No, the median is the value where half the data is less than or equal to that value and half the data is greater than or equal to that value.

So if your ordinal scale had 100 respondents then find the value that has at least 50 less or equal and 50 greater than or equal. It would only be 3 if half the responses were to either side. If 1 person said 1, 2 people said 2, 3 said 3, 4 said 4, and the remaining 90 said 5, then 5 would be the median.

The median works when the data is ordered, but would not make sense for nominal/unordered data, like what is your favorite color?

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  • $\begingroup$ @ Greg, Thank you for your reply. What are the cases when the median cannot be used with ordinal level data? $\endgroup$ – Anne May 25 '11 at 3:24
  • $\begingroup$ I cannot think of any cases of ordinal data that you cannot use the median, there are probably cases where other summaries are more meaningful (small number of levels, just give graph or count of everything). $\endgroup$ – Greg Snow May 25 '11 at 15:41

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