The answer to your question depends on what statistical approach you need to adopt to analyse your data to answer your questions. For example, if you use some kind of a linear model, then it is the residuals, not the data per se, that need to fulfill the normality (and constancy of residual variance) assumptions. 

Your description is a of likert-type scale, rather than a true likert scale.  The true likert scale is a multi-item scale whereas your question suggests you have a single item. Each item in the true likert scale is ordinal data and the intervals are not known for certain. 

However, if you have multiple items of this type, they can be combined into a true likert scale (e.g. through totalling four or five items). The resulting distribution is then assumed to be metric and can be tested for normality. So four items each with a range of 1 to 5 would combine together to yield a likert scale with a range from 4 to 20, with combined data that is generally treated as being metric.  That is one of the points of having multi-item likert scales - that they convert these rough ordinal measures into metric data more suitable for multivariate analysis. 

With just a single item, you might still argue that you have interval data, but unless you can test this with an appropriate method (e.g. Rasch modelling) that is a dangerous assumption to make. For example, your respondents may interpret the 'neutral' code as also being a 'don't know' and this may be closer to the "not important" answer than the "important" answer, making the intervals around the midpoint asymmetric, and thus violating the assumption of equal intervals between data points that is required for metric data suitable for regression.

I do agree with your other correspondent above - that from the point of view of the confidence intervals in a regression it is the distribution of the residuals that matters.  But from the point of view of theory, the assumption that your single item scale has a linear relationship with another variable is difficult to support for the reasons I have canvassed above. Any test of normality of the original variable is going to be pretty meaningless for the same reasons.

I would recommend using statistical methods that only rely on the assumptions of ordinal rather than metric data, such as Spearman's rank correlation coefficient. Or, If you are trying to predict this ordinal item, you could use ordinal regression. Alternatively, if you are using this ordinal item as a predictor,  you could convert it into dummy variables. More generally, there are plenty of nonparametric methods available that do not rely on the strict assumptions of regression and may be more suitable for your data. 

I hope this is of some help.

Dear Pooneh

agree with others that it depends on your statistical model you attend to use, but you must be aware  that you have four groups here so if you want to use ANOVA for example the four groups must be normaly disributed the thing I doubt in case of small sample size as (10, 17). So I suggest to do as Malcolm advised and use a  statistical model which does not assume normal distribution.

good luck

I'm afraid that there is some misunderstanding. Using the Likert scale(1 to 5) for survey, you are in the area of so-called non-parametric statistics with the Ordinal scale of data. All kinds of analyses within non-parametric statistics are distribution free. So the question about normality testing is meaningless. BTW, you can't use any ANOVA for these data.

Dear Mr. Grabov, can you provide a reference for your statement that ordinal data is distribution free? It sounds very interesting, do you also suggest that the data can be free of normality test as well? Thanks a lot!

I am working on a similar issue, where I have matrix, and each cell in the matrix has a Likert scale scoring. I am trying to compare any cell with another. Perhaps ANOVA is the method to do so. But I am not sure, if for this ordinal scale of data, how to check its normality and proceed for ANOVA. 

Can someone please help me on this. Thanks.

Hi, sorry for the delay with ny response due to abnormal pressure. There is an interesting paper on this issue: SHIKEN: JALT Testing & Evaluation SIG Newsletter. March 2011. 15(1) 10-14

About scales of measurement - see: https://en.wikipedia.org/wiki/Level_of_measurement

Hello, Clason and Dormody (1994) also have a say about the non-normality of single likert items « It is difficult to see how normally distributed data can arise in a single Likert-type item. The data will frequently be skewed, and often these items do not capture the true limits of the attitude. » 

https://pdfs.semanticscholar.org/e2ad/36059a7894c89ea566811eb82ad92acd88a7.pdf

I agree with Pavel Grabov that ordinal data can only be analysed with nonparametric statistics; ordinal data do not satisfy any of the requirements of parametric statistics which includes normality of the data. It is therefore not appropriate to determine the normality of ordinal data.

Please provide some references for non-normal nature of Likert scale data.

Article LIkert scales, levels of measurement adn the “laws” of statistics

Likert scales, levels of measurement and the “laws” of statistics

by G. Norman (2010)