# Analysis of Variance ANOVA

T-tests or z-tests can be only performed for comparisons of a maximum of two samples/populations. However, when more than two samples/populations are compared, simple t-tests or z-tests are not enough. While analysis of variance (ANOVA) has already been performed when t-tests or z-tests were conducted for mean analyses in the earlier module. For example, the t-test/z-test formulas contain the variance (s2 or σ2), which indicates that analysis of variance, ANOVA has already been conducted without mentioning it in the earlier t-test/z-test. In general, the analysis of variance, ANOVA is performed when the total number of comparisons increases over two. Like the t-test and z-test, many statistical methods are available for different experimental situations. The most basic method is the single-factor analysis of variance, which is also known as the one-way ANOVA simply because this method contains just one factor (single factor). A single factor with a maximum of two levels can still be analyzed using the t-test or z-test or other appropriate tests. However, the single factor with more than two levels will need ANOVA with advanced methods depending on the experimental situations. The most basic single factor with more than two levels is the completely randomized design (CRD). In a completely randomized design, experimental units are randomly assigned to all levels of the factor. Any experimental unit has an equal chance/probability of getting selected for any level of the factor. Therefore, the detail name for the completely randomized design can be completely randomized equal replication design, or completely randomized equal replication experiment due to the equal probability for any experimental unit to be selected for any levels of the factor (Hinkelmann and Kempthorne 2008).

Assume that three different fuel types are tested for fuel economy with respect to the miles per gallon. Assume that the experimental units are randomly selected from a homogenous group of vehicles considered as identical twins with respect to the experimental conditions. Any experimental unit (e.g. test vehicle) will have an equal chance of getting any of the fuel types (any of the treatments or treatment combinations is the statistical jargon). It is also noted that each experiment is run in the exact same experimental conditions. Therefore, the experiment in this fuel type study will be considered as the completely randomized design (CRD). The fuel economy in miles per gallon here is known as the responses or dependent variables while the fuel types are independent variable or factor or explanatory variable or predictor variable.