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I understand that if a graph is close to the shape of a bell shape (and the mean tends to zero, and the SD is close to 1), then we can say it is normally distributed. Visually inspecting a normal curve can be subjective as well. How should I confirm if the residuals of the curve below is normally or approximately normally distributed? I am using this judgment to as a pre-determination of ANOVA assumptions.

enter image description here

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  • $\begingroup$ Can you give some context, why do you need an approximate normal distribution? How to judge the approximation depends on purpose. $\endgroup$ Sep 24, 2017 at 18:38
  • $\begingroup$ I am inspecting my data to ensure that they do not violate the assumptions for conducting an ANOVA and a Regression Analysis $\endgroup$
    – Vyas
    Sep 24, 2017 at 18:41
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    $\begingroup$ "if a graph is close to the shape of a bell shape (and the mean tends to zero, and the SD is close to 1), then we can say it is normally distributed" -- (i) the mean and standard deviation are irrelevant to whether or not it's normally distributed; (ii) many things that are very much not normal may look more or less bell-shaped, so merely looking bell shaped is not a means to assert normality; (iii) that doesn't look at all bell-shaped anyway; it's clearly bimodal. What's your response? (iv) are those residuals from the actual model you want to assess the assumptions of? $\endgroup$
    – Glen_b
    Sep 25, 2017 at 4:48
  • $\begingroup$ Consider trying to identify the two groups that seem to have accounted for two distributions here. You might conduct a separate analysis on each group. $\endgroup$
    – rolando2
    Sep 25, 2017 at 15:46
  • $\begingroup$ What do you mean by two groups? Why do you say it is bimodal? I see the term "conservative approach" is used. If the normality test says that the distribution is not normal, how should I proceed to conduct the ANOVA and Regression, apart from reporting that the sample is not normally distributed? $\endgroup$
    – Vyas
    Sep 26, 2017 at 2:47

1 Answer 1

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As you said, visually inspecting is really subjective, but in your example, the histogram indicates two modes and the part with the most theoretical probability density (near 0) has too few points, so you have a strong "intuitive evidence" that a normal distribution assumption might not be appropriate.

Another (also subjective) form of inspecting normality is by looking at the Q-Q Plot. On the other hand, you can use a Normality Test to decide if you consider the normality assumption is valid with only the significance level as a subjective factor.

Keep in mind that, as commented, you should understand and considerate the consequences of your normality assumption, like what parts of your analysis can be invalid because of it.

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