The Analysis of Variance (ANOVA) test involves calculating several different values and using them to determine whether there are significant differences between groups. Here are the steps involved in calculating ANOVA:

Step 1: Calculate the mean of each group. Calculate the mean of each group by summing the values of each observation in a group and dividing by the number of observations in that group.

Step 2: Calculate the overall mean. Calculate the overall mean by adding up all the observations in all groups and dividing by the total number of observations.

Step 3: Calculate the sum of squares between groups. The sum of squares between groups represents the variability between the groups. To calculate it, subtract the overall mean from each group mean, square the result, and then sum up all the squared differences.

SS_between = Σ (n_i (x̄_i – x̄)^2)

where n_i is the number of observations in group i, x̄_i is the mean of group i, x̄ is the overall mean.

Step 4: Calculate the sum of squares within groups. The sum of squares within groups represents the variability within the groups. To calculate it, subtract each observation from its respective group mean, square the result, and then sum up all the squared differences.

SS_within = Σ Σ (x_ij – x̄_i)^2

where x_ij is the jth observation in group i, x̄_i is the mean of group i.

Step 5: Calculate the degrees of freedom. The degrees of freedom represent the number of independent pieces of information used in the calculation of each sum of squares. The degrees of freedom for the between-groups sum of squares is one less than the number of groups, while the degrees of freedom for the within-groups sum of squares is equal to the total number of observations minus the number of groups.

df_between = k – 1 df_within = N – k

where k is the number of groups, N is the total number of observations.

Step 6: Calculate the mean square values. The mean square values are calculated by dividing the sum of squares by the degrees of freedom.

MS_between = SS_between / df_between MS_within = SS_within / df_within

Step 7: Calculate the F-ratio. The F-ratio is calculated by dividing the mean square between groups by the mean square within groups.

F = MS_between / MS_within

Step 8: Determine the p-value. The p-value is the probability of obtaining an F-ratio as extreme as the one calculated, assuming that there is no difference between the groups. The p-value can be obtained from a statistical table or calculated using software.

Step 9: Interpret the results. If the p-value is less than a pre-determined significance level (such as 0.05), it indicates that there is a significant difference between at least two of the groups. If the p-value is greater than the significance level, it indicates that there is not enough evidence to conclude that there are significant differences between the groups.

In summary, the ANOVA test involves calculating several values, including the sum of squares between groups, the sum of squares within groups, the mean square values, the F-ratio, and the p-value. These values are used to determine whether there are significant differences between groups.