Shapiro-Wilk Test vs Boxplots

I have a question with regards to normality of the data. When I do a shapiro-Wilk test I'm getting a p>0.05 (which means that my data is normally distributed). However, when I plot my data using boxplots, the median is not centered in the middle and seems skewed.

I'm not sure what to believe? How do I decide if my data is normally distributed or not? Can I go ahead and assume its normally distributed because the shapiro-Wilk test confirms that quantitatively? Would I be criticized for making that assumption if my boxplots look skewed?

• Some issues suggested by this post, which you therefore might wish to explore further, are (1) $p\gt 0.05$ is the opposite of "significant." (2) Boxplots are not terribly useful for assessing Normality. (3) No hypothesis test, such as the S-W, "confirms" an assertion: at best it can show the assertion is consistent with the data (given certain assumptions).
– whuber
Dec 16 '20 at 22:01
• Yes, my bad. I meant that I'm getting a p>0.05. So, what I'm understanding is that the S-W test is not enough to assume normally distributed data? Dec 16 '20 at 22:11
• It depends on why you are checking the data distribution. In many (perhaps the great majority) of situations where people come here asking about normality tests, it turns out they don't need those tests in the first place. When they do need them, it's usually to assess whether some other statistical procedure might be appropriate. In those cases what almost always matters is not whether the data depart from looking normal, but how they depart from normality. For this reason, many people begin with a graphical assessment of the data, such as a Normal QQ plot.
– whuber
Dec 16 '20 at 22:13
• Ah I see. I'm checking normality for deciding whether to use a t-test or a non-parametric mann-whitney test. But my sample size is small (its a total of 25 participants), and when I look at the qq-plot, it sometimes hovers close to the line but not full on it. Its hard to tell, which is why I wanted a quantitative value to decide whether it is normally distributed or not Dec 16 '20 at 23:00
• @whuber I added what the qqplot looks like.. im not sure what to make of this. Is it normally distributed? Its not fully on the line but its also not very skewed Dec 16 '20 at 23:13

1. No data are ever truly normally distributed. The best you can get is that the data are compatible with the normal distribution, i.e., look like at least fairly typical normal data. But they are not really normal anyway (because for starters no real data are truly continuous or have a truly infinite value range).

2. Shapiro-Wilks $$p>0.05$$ does not mean that the data are normally distributed, it only means truly normally distributed data could look like this.

3. I don't know what you need the normal assumption for, but usually in situations where this assumption is "officially" required, data looking approximately normal will be fine (as long as there are no issues with other assumptions).

4. For interpreting your QQ-plot (which is not a boxplot by the way), you need to take into account that there is random variation, and even if your data were truly normal, they wouldn't look like a straight line. In fact your sample size seems small and the pattern you see there doesn't look suspiciously non-normal at all for the small sample size that you have.

5. Therefore my impression is that there is no contradiction at all between the QQ-plot and the Shapiro-Wilks $$p>0.05$$ - both say that your data look pretty normal.

• Ah okay, that was very helpful. Thank you! Dec 16 '20 at 23:50