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I would like to run one-way ANOVA test on my data. I saw that one of several assumptions for one-way ANOVA is that there needs to be homogeneity of variances. I have run the test for different data-sets. I find sometimes my p-values are larger than 0.05 and for some datasets it is smaller.

As I understand, if the p-value is smaller than 0.05, then I can reject the null hypothesis and tell that the variances are not equal (and then it makes sense to use ANOVA?)

The problem is that I see in different sources diferent conclusions: in some they run ANOVA if they reject the null hypothesis, for some they run ANOVA if they cannot reject the null hypotheis. For example,

here it says that if is smaller we can run the ANOVA: enter image description here

But here it recommended that p-value>0.05

To me, intuitively, it sounds like p-value should be greater than 0.05 but I would like to get clear answer regarding that.

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"As I understand, if the p-value is smaller than 0.05, then I can reject the null hypothesis and tell that the variances are not equal (and then it makes sense to use ANOVA." Your understanding is incorrect. As a measure of goodness of fitness of data, null hypothesis has to be accepted if alpha is less than or = .05. If you want to perform statistical test of variability across groups, you need an implementation of Chi-square test of variability across groups. Your intuition (question in last part of body) -p-value should be greater than 0.05 is not logical. This value indicates sampling fluctuations or error arising on account of chance in sample selection. An (sampling ) error greater than 5% is not acceptable by any standards.

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