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I have read that the t-test is used when the population is normally distributed. How can I determine if my data are normal given that I am using 5-point Likert scale with a sample size of 100? What does standard deviation describe?

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How to judge if 5 point likert scale data are normal distributed?

Values on 5-point ordinal scales are never normally distributed. But that's probably not the question you really need answered.

I have read that the t-test is used when the population is normally distributed.

It's an assumption of the test, but it's often reasonably robust to mild deviations from the normality assumption, especially if the distribution is nearly symmetric.

How can I determine if my data are normal given that I am using 5-point likert scale with a sample size of 100?

You data are not normal. But real data are likely never actually normal. The useful question is not "are my data normal" (no, they're not), but something more like "is the extent to which my data deviate from normality enough to affect my inference in ways I need to worry about?".

To answer that is something that changes from context to context, and person to person. It's not something about which absolute prescriptive statements can be made -- but the size and kind of impact on inference (effect on significance level and power under various alternatives) can be assessed.

What does standard deviation describe?

Basically, how far a typical* observation is from the mean.

* (in a particular sense of the word 'typical', and a particular way of measuring 'how far')

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Normal distribution is a continuous distribution while 5-point Likert-type scale is an ordinal variable, so by definition it is not normally distributed.

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  • $\begingroup$ This is true, but you still might wonder if it's "normal enough" to proceed with various analyses. $\endgroup$ – Andrew M Dec 6 '14 at 2:17
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    $\begingroup$ Tim, I don't agree with your superficial statement. Whether a rating scale (Likert, Osgood, Stapp or whatever) of whatever length of 3+ length is ordinal or interval - is the subject of faith. Until you have real underlying continuous data to judge it. Also, any measurement scale is not continuos, so what? Does it mean it cannot approximate a continuous distribution? It's true that a rating scale cannot be normal just because it's discrete, but it's trivial. It nevetherless can be seen as discrete approximation. $\endgroup$ – ttnphns Dec 6 '14 at 9:52
  • $\begingroup$ Yes it is a subject of faith. But there are two good arguments: (1) it is discrete, that's fact, (2) if you consider it as approximation of continuous scale you also have to assume that people use different categories on the scale in a linear way - while in many cases that is not the case. Why not just use methods that will make this kind of data continuous, e.g. IRT? $\endgroup$ – Tim Dec 6 '14 at 14:04
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If you are considering t-tests on Likert items, I would primarily be worried about how many 1's and 5's there are, since those values might represent censoring of responses that could exceed 1 or 5 if it permitted. This censoring is much problematic than the fact that you would be treating a discrete distribution as if it were continuous. See How to model this odd-shaped distribution (almost a reverse-J) for a good overview for what can go wrong if you use typical approaches on censored data.

In any case, Group differences on a five point Likert item has some excellent answers for approaches to testing for group differences on likert items and the drawbacks to treating bounded, ordinal data as if it were continuous.

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@Tim is correct, likert data cannot be normally distributed. Likert data are discrete and bounded, normal data go to infinity in both directions and can take any value in between.

The answer to your other question is that the standard deviation means pretty much the same thing whether your data are likert-type, normal or something else. The standard deviation is, essentially, how far away from the mean your data are. When data are normally distributed, it is also true that, e.g., 68% of your data will be within +/- 1SD, but that won't necessarily be true for non-normal data. When you have likert data and you want to think about how much your data vary, you may prefer a non-parametric measure like the interquartile range.

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I would suggest to give the data a hard look via a diagram. That sounds unscientific, but eyeballing is a really powerful tool for lots of problems. Assessing normality of a Likert scale is one of them. Look at the distribution and imagine you draw a normal distribution: would the data fit into the curve, or would there be gross violations, e. g. are the values shifted to the left or right?

Alternatively, there are things like a Kolmogorov-Smirnov test for normality, but with a lot of cases that detects non-normality 10 times out of ten.

And yes: it's not normally distributed because it's not a continuous variable. But it's a very standard method of measurement in a lot of fields and it's often treated like a metric variable because all other choices are even worse. Sorry, I'm a little bit miffed about this problem because it comes up in my work every week.

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I think you have two questions here:

  • how do you describe the distribution of a set of Likert scores? (by your question is such a set normally distributed)

  • how do you tell if two sets of Likert score are 'different' (or one Likert score different from the one that is most 'normal'?

For the first one, only continuous data can be normally distributed, and a Likert score is discrete, 1 out of 5 values. (you can fake it sort of by treating 1-5 a real values but that's not the point here. So the appropriate analogous distribution is a binomial distribution on 5 items (for answers 1,2,3,4,5 having probabilities of 1/16, 4/16, 6/16, 4/16, and 1/16 respectively). and the analogous title question should be:

How to judge if 5 point Likert scale data are distributed like B(4)?

(4 = 5-1 which is just how it works out).

To the second question, you want to see if a given set of Likert data is 'like' B(4) or like another set. Here I would use Chi-squared on the difference of 1's, 2's, 3's, 4's and 5's.

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If you have ordinal data why would you be concerned about a normal distribution? The only reason I can think of is if you are thinking of a latent trait that is manifested categorically; in that case, one can make the assumption that the latent trait is normally distributed. If you consider it a latent trait, and are truly concerned about the normality of that latent trait, you may be able to use an ordinal probit regression (which assumes a latent normal variable) and assess the fit (though a smarter person than I may have a better way of doing that).

Otherwise, if you are not thinking of the latent trait interpretation, don't do a t-test or z-test. Use non-parametric tests, Chi2 test, or an ordinal logit/probit. If you do an ordinal logit (so you can include multiple predictors), test the parallel lines assumption (see this implementation in Stata using Brant's test, Problem 2 on Page 7). Interpreting ordinal scales as continuous is done all the time in certain fields - and in my experience often statistical significance tests treating it as continuous vs tests treating it as categorical don't come out much different. That said, I'm of the mindset that if appropriate tests are available for your data, and their implementation is feasible, you should use them.

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