It is important to note that ANOVA is not robust to violations to the assumption of independence. This is to say, that even if you violate the assumptions of homogeneity or normality, you can conduct the test and basically trust the findings. However, the results of the ANOVA are invalid if the independence assumption is violated. In general, with violations of homogeneity the analysis is considered robust if you have equal sized groups. With violations of normality, continuing with the ANOVA is generally ok if you have a large sample size .

The T-test tutorial page provides a good background for understanding ANOVA ("Analysis of Variance"). Like the two-sample t-test, ANOVA lets us test hypotheses about the mean (average) of a dependent variable across different groups.
While the t-test is used to compare the means between two groups, ANOVA is used to compare means between 3 or more groups.
There are several varieties of ANOVA, such as one-factor (or one-way) ANOVA, two-factor (or two-way) ANOVA, and so on, and also repeated measures ANOVA. The factors are the independent variables, each of which must be measured on a categorical scale - that is, levels of the independent variable must define separate groups.
One-Way ANOVA Example
One-factor ANOVA, also called one-way ANOVA is used when the study involves 3 or more levels of a single independent variable. For example we might look at average test scores for students exposed to one of three different teaching techniques (three levels of a single independent variable).
ANOVA Statistics
The null hypothesis for ANOVA is that the mean (average value of the dependent variable) is the same for all groups. The alternative or research hypothesis is that the average is not the same for all groups.
The ANOVA test procedure produces an F-statistic, which is used to calculate the p-value. As described in the topic on
Statistical Data Analysis if p < .05, we reject the null hypothesis. We can then conclude that the average of the dependent variable is not the same for all groups.
With ANOVA, if the null hypothesis is rejected, then all we know is that at least 2 groups are different from each other. In order to determine which groups are different from which, post-hoc t-tests are performed using some form of correction (such as the Bonferroni correction) to adjust for an inflated probability of a Type I error.
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Hello Charles,

Thank you very much for your reply!

1) For experiment 1, both data sets that failed the normality test (p= and p=) are not symmetric, according to the box plot. Therefore, a nonparametric test should be used for the analysis, right?

2) For experiment 2, there are two experimental groups. I only have three values for each group. The data for group A are: , , (normality test P<). The data for group B are: , , (normality test P=). The results from t-test (p=) and Mann-Whitney Rank sum test (p=) are very different.

Thank you!