Final answer:
ANOVA is quite robust to deviations from normality/variance assumptions when the sample sizes are large, because the Central Limit Theorem ensures that large sample means are approximately normally distributed.
Step-by-step explanation:
The Analysis of Variance (ANOVA) is quite robust to deviations from normality/variance assumptions when sample sizes are large. This is because, as sample size increases, the Central Limit Theorem assures that the sample means will be approximately normally distributed, regardless of the distribution of the data. Consequently, the importance of the original data's distribution diminishes with larger sample sizes, allowing the ANOVA to produce reliable results despite deviations from normality.
Large sample sizes therefore contribute to the robustness of ANOVA when populations do not meet the assumption of normal distribution or when there is heterogeneity of variances. ANOVA can be sensitive to deviations from these assumptions with small sample sizes, and homogeneity of variance is an essential prerequisite for the proper application of ANOVA. If the assumption of normality is violated with small samples, additional statistical testing may be needed to justify the use of ANOVA.
However, in practice, when samples are sufficiently large (often considered as greater than 30), the distribution of sample means can be reasonably approximated by a normal distribution, thus making a one-way ANOVA less sensitive to violations of normality. The F Distribution is used to conduct hypothesis tests in a one-way ANOVA framework.