# Normality test using normal Q-Q plot and histogram

I have plotted a normal Q-Q Plot and a histogram to check the normality of this set of discrete data. My interpretation is the data are not normally distributed since they do not fall on the linear line and the histogram does not shape like a bell curve. However, for discrete data which contains lots of repeated value and the data value only varies from 1 to 7, is my interpretation correct?

The numerical test result is as follow: Asymptotic one-sample Kolmogorov-Smirnov test

D = 0.16234, p-value = 0.03455

• Checking normality is unlikely to be meaningful or useful here. Check out some of our many threads on this topic.
– whuber
Jan 11 at 13:57
• whuber anticipated my comment! Checking normality for a Likert scale doesn't make sense. Jan 11 at 14:00

While I agree with whuber's comment that this is unlikely to be useful, and while you don't say why you are doing this, I think that, nevertheless, something useful can be said.

Of course, if you have Likert scale data, or any other discrete data, you know in advance that it isn't normally distributed. So a formal test of this is even less likely than usual to be useful (and that's saying something!). Rather, we can ask how close to normal the data are. For this, and in particular you could look at how close to the middle of the data the line in the QQ plot goes. For me, it looks pretty close, but it's clearly off by a bit, showing a bit of left skew. The histogram confirms this.

If you wanted to do something more, as an academic exercise, you could try adding a bit of random noise to the data, so that it becomes continuous, and then do another quantile normal plot.

Why do this? Again, I don't know. But playing around can be fun, and maybe you have something more specific in mind.

• (+1) It's hard for beginners to know quite what makes sense. A normal distribution is defined over the entire real line, negative, zero, and positive values alike, but beginners are expected to see that such a definition doesn't rule out a normal plot being useful for adult heights, which can look close to normal. So, it's perhaps not so obvious that Likert scales should not be compared with normal distributions. While e.g. a formal test is indeed pointless, the graph need not be. I've a small collection of authors pointing out that normal quantile plots can be useful for any numeric variable. Jan 11 at 15:10
• The argument runs like this. First, normal being a reference distribution does not commit us to thinking that it will fit. Second, a normal quantile plot lets us see level, spread, skewness, outliers, granularity, gaps and much else. Which other plots are so versatile? Jan 11 at 15:13

I will disagree (simply because I can) that the data necessarily will be non-normal. You can get "normal" data in some respects (a bell-curved histogram with many 4-values on a scale of 1 to 7). But I agree with the others that

1. This is not typical of Likert scales (see below reference).
2. Its importance is probably not as major as some sources claim.

As Peter noted, it would be useful to know what you are doing with this data. That would guide decision-making on what you should do. There exist plenty of non-parametric methods if this is indeed a problem, but that would be determined largely by the context of the analysis. Parametric tests often do well even with this assumption violated.

#### Reference

Sullivan, G. M., & Artino, A. R. (2013). Analyzing and interpreting data from likert-type scales. Journal of Graduate Medical Education, 5(4), 541–542. https://doi.org/10.4300/JGME-5-4-18

• You can certainly get something that's sort of bell shaped, but the Normal distribution is continuous. Jan 11 at 17:47
• Certainly. I think that is actually a point that is subtly important but strangely missing from most psych stats texts I have seen. Jan 11 at 18:03
• Being continuous is a matter of principle. In practice even so-called continuous measurements come with some convention about resolution (sometimes called precision). People's heights are typically quoted to say cm; even if mm were to be used, that would be a discrete scale too. I am not expert or trained in psychology but suggest that a convention of reporting IQ as integers in no sense rules out comparison of data with normal or any other distribution devised for continuous variables. Jan 12 at 14:38