# Question about Normality of 5-point Likert scale

I'm really having a hard time trying to normalize my data. My study is about green IT product acceptance. There are eight factors, six are 5-point Likert scale (Str.disagree - Str.agree), one is 2-point Yes/No question, and the other one is 3-point scale (Don't know/Maybe/Know). I have 618 returned questionnaires.

I add data into SPSS, compute variables, and round them up. I explored skewness and kurtosis of all factors and found that they're not normally distributed. I tried a Two-Step Transformation to Normality and Box-Cox, but the KS and SW Sig. are still .000. However, Q-Q plot change from loosely distributed to perfectly follows the slope.

My questions are:

1. Is it a crucial problem if my data isn't normally distributed?
2. Someone said Likert scales are not normal in their nature, is that true?
3. Do you know any literature to cite and explain why Likert scale isn't normally distributed?
4. Do I need to normalize all variables before I do Structural Equation Modeling?
• Note that a number of questions on site deal with some aspects of your question. · · · · Likert scales are actually the things constructed from Likert items. Likert items (individual components on that ordinal scale) are clearly discrete - they only take a few different values. There's no reasonable way to claim the categorical response on a Likert item is itself drawn from a normal distribution, even if you assume the ordinal categories are equi-spaced ("interval" on Stevens' typology of scale). – Glen_b -Reinstate Monica Oct 28 '16 at 4:31
• Note that even if skewness and excess kurtosis were both 0 they still wouldn't be normally distributed (distinctly non-normal distributions can have the same skewness and kurtosis as the normal distribution). Transformation isn't a solution -- transformation won't make discrete things continuous. A scale constructed from many such Likert items might sometimes be argued to be reasonably approximated by a normal distribution -- but it will never actually be normal (not that the distinction will necessarily matter). – Glen_b -Reinstate Monica Oct 28 '16 at 4:31
• Testing normality is not much use either, since you'll reject when $n$ is sufficiently large even if it's perfectly adequate to treat it as if it were normal; similarly you can fail to reject at small sample sizes even when there's consequential deviations from normality in the population. You're dealing with an effect-size type problem, not an issue related to statistical significance. · · · · · · Please consider clarifying questions 1 and 4 to indicate which specific problem and phenomenon you refer to respectively (you discuss several things in your Q). – Glen_b -Reinstate Monica Oct 28 '16 at 4:33
• Another possibility is to treat the items like the categorical indicators they are and use SEM with an ordinal logit or probit link function. – Noah Aug 30 '17 at 4:49

1. Is it a crucial problem if my data isn't normally distributed?

Data are (almost) never normal. Whether that's an issue depends what forms of deviation from normality the procedure you want to use is sensitive to (and how much), how non-normal it is and in what way it's non-normal (strictly we're talking about the distribution the sample was drawn from rather than the sample itself).

Where there's much doubt about the potential impact, try to avoid assuming things you don't need to.

1. Someone said Likert scales are not normal in their nature, is that true?

They're discrete and bounded, normal distributions are not.

1. Do you know any literature to cite and explain why Likert scale isn't normally distributed?

"They're discrete and bounded, normal distributions are not." is 100% of the literature you should need ... it's simple mathematical fact, instantly apparent from looking at the definition of the normal density and a Likert scale that anyone could check for themselves. Would you really quote literature to support a plain statement of fact like "3 is not an even number"?

1. Do I need to normalize all variables before I do Structural Equation Modeling?

What is assumed normal in SEM and does that imply marginal normality of every variable? (My understanding is that the errors might be assumed MVN but that doesn't imply marginal normality of all variables, so it wouldn't automatically suggest transformation of variables that don't look normal on their own.

Note that "normalize" carries a more common particular meaning (or actually a couple) distinct from "transform to normal distributions" (see the tag wiki for for a brief explanation)