# "Intervals" for a scale-based answer

I have to analyse data that look at the relevance of some posts. Basically each post is rated by 100 persons, and the scale looks something like this (like a Likert scale):

1. Irrelevant
2. Slightly relevant
3. Relevant
4. Useful
5. Extremely Useful

But each person can choose a certain position with a 0.25 increment on this scale. So, more specifically, the scale looks like this [1, 1.25, 1.50, 1.75, 2, 2.25,...3,...4,...,4.75,5]

I was asked to compute some intervals for each post. My first reaction was to compute the mean and the 95% CI, which is evidently wrong since the data is not normally distributed. Which is the best approach for this problem, where should I start looking? I read something about Bayesian Approach on this site, will that do? Thanks

You correctly note that you cannot use "usual" formulas to calculate confidence intervals for the mean, since your data is not normally distributed. One possibility would be bootstrapping the mean and extracting bootstrap confidence intervals.

However, I would argue that confidence intervals for the mean are not necessarily best to describe your data. Instead, I suggest you look at simple quantiles. For instance, a given post may be rated 4.5 on average, and the 25% quantile may be 3.5 (that is, 25% of respondents rated lower than 3.5), while the 75% quantile may be 4.8 (that is, 25% of respondents rated higher than 4.8). Thus, you could say that 50% (75%-25%) rated it between 3.5 and 4.8.

Could you clarify what question you are trying to answer?

• The quantiles answer a substantially different question than confidence intervals do, so before proposing such an answer, you should seek clarification from the O.P. that this is really what they need, rather than what they actually asked for.
– whuber
Jan 7, 2013 at 14:27
• The OP "was asked to compute some intervals for each post" and asks for "the best approach to this problem". I honestly believe that quantiles are the best approach for the problem as it is described. If I misunderstood the OP's question, I'll happily change the answer or delete it as necessary. Jan 7, 2013 at 14:43
• I am making three points, Stephen. First, the quantiles and confidence intervals answer different questions. Second, their values will be substantially different numerically, so the choice matters. Third, because of this it is crucial to be clear about what question you are answering. It is not correct to substitute empirical quantiles for a confidence interval of a mean and it is (at best) misleading to interpret "some intervals" as any intervals you choose (when the title itself specifies that they should be "confidence" intervals).
– whuber
Jan 7, 2013 at 14:53
• I certainly agree that the mean and CIs will differ markedly from quantiles. I edited my answer. Jan 7, 2013 at 15:04
• I edited my question a bit, thank you very much for your answers.I don't know exactly what intervals I am supposed to compute, i was told just 'intervals' so far, so.. i assumed it with be confidence intervals. Still, the quantiles range (I took the interval between 75% and 50% quantile ) seem to make a lot of sense with the data. I think the request has to do more with describing the data that I have , rather than making inferences about the whole population, so I will stick with this for now. Thanks. Jan 7, 2013 at 16:35