Is testing for deviation from the Guassian distribution pointless?

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Kvothe the Arcane
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One can use a test like the Shapiro Wilks to see whether a particular distribution is or is not normally distributed. It tests the null hypothesis that the sample comes from a normal distrbiution. And if p<0.05, you fail to reject the null hypothesis. As a result, tests like One way ANOVA cannot be used as they require normally distributed data.

But it seems to me that even slight deviations from the Guassian curve, particularly with large n, result in a low p. And a discrepancy arises between what might be interpreted from a histogram or qqnorm/qqline plot as normally distrbuted and the q value.

Is testing numerically then useless?
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Gregorius
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(Original post by Kvothe the Arcane)
One can use a test like the Shapiro Wilks to see whether a particular distribution is or is not normally distributed. It tests the null hypothesis that the sample comes from a normal distrbiution. And if p<0.05, you fail to reject the null hypothesis. As a result, tests like One way ANOVA cannot be used as they require normally distributed data.
I think you mean p < 0.05 leads to a rejection of the null :-)

But it seems to me that even slight deviations from the Guassian curve, particularly with large n, result in a low p. And a discrepancy arises between what might be interpreted from a histogram or qqnorm/qqline plot as normally distrbuted and the q value.

Is testing numerically then useless?
Yes, this is a problem with tests for any distributional assumptions. A large sample size will almost inevitably lead to a rejection of the null - whereas what you really want to know is the way in which the distribution differs from the normal and the consequences (if any) for any subsequent tests like ANOVA or ANCOVA.

There are two approaches that we tend to take IRL: the first is to inspect qqnorm/qqline plots to see the form of the deviation - if it's not too horrible, then subsequent tests (especially those built on the t-distribution) tend to be quite resilient to minor deviations from normality - and we proceed with gay abandon. The second approach is just to use the bootstrapped version of the subsequent test - provided there are no problems of heteroskedasticity.
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