Describe, in your own words, the following terms and give an example of each.
a. Analysis of Varianceb. one-way ANOVA c. between-group variance d. within-group variance
e. sum of squares between groups f. sum of squares within groups g. post-hoc analysis h. F- distribution i. two-way ANOVA

(a) ANOVA is a general technique that can be used to test the hypothesis that the means among two or more groups are equal, under the assumption that the sampled populations are normally distributed.
Example: The production of three varieties of wheat sown in 12 plots is recorded and ANOVA is used to test if there are significant differences in the production of the three varieties.
(b)In one way ANOVA, the data are classified according to only one criterion.
Example: A Grammar test is given to 5 students each sampled from 4 schools to assess the significance of possible variation in performance.
(c) This measures the contribution of both different treatments and chance to inter-group variability. For calculating this, we take the total of the sum of squares of the deviations of the means of the different groups from the grand mean and divide this total by the degrees of freedom.
(d) This measures those inter-group differences that are due to chance only. For calculating TSE, we take the total of the sum of squares of the deviations of the items from the mean values of the respective samples and divide this total by the degrees of freedom.
(e) It is denoted by SSC. It is the sum of squares of the deviations of the means of the different groups from the grand mean.
(f) It is denoted by SSE or TSE. It is the sum of squares of the deviations of the items from the mean values of the respective samples.
Examples for (c), (d), (e) and (f):
A researcher wishes to determine whether there is a difference in the average age of elementary school, high school, and community college teachers. Teachers are randomly selected. Their ages are recorded below. Find the critical value F0 to test the claim that there is no difference in the average age of each group. Use a = 0.01. Elementary Teachers: 23,28,27,25,37,52. High School Teachers: 36,41,38,47,42,31. Community College Teachers: 39,45,36,61,45,35.
SUMMARY
Groups / Count / Sum / Average / Variance
ELT / 6 / 192 / 32 / 119.2
HST / 6 / 235 / 39.16667 / 30.16667
CCT / 6 / 261 / 43.5 / 91.9
ANOVA
Source of Variation / SS / df / MS / F / P-value / F crit
Between Groups / 404.7778 / 2 / 202.3889 / 2.516579 / 0.114173 / 6.358873
Within Groups / 1206.333 / 15 / 80.42222
Total / 1611.111 / 17
(g)Post-hoc analysis refers to looking at the data - after the experiment/statistical analysis has concluded - for patterns that were not specified before-hand. The analysis may also involve digging deeper into the statistical results already obtained.
For example, in an ANOVA involving more than two treatments, if a significant main effect or interaction is found, then we can only conclude that there is a significant difference amongst the levels of treatments somewhere. But we still have to isolate exactly where the significant differences lie. This is done usually using Tukey's HSD post- hoc test.
(h) The F-distribution comes into picture relevant when we try to calculate the ratios of variances of normally distributed statistics. Let there be two samples with n1 and n2 observations, thenthe ratioF = s1^2 / s2^2 (where s1^2 and s2^2 are the sample variances) are distributed according to an F- distribution with df1 = n1 - 1 numerator dof and df2 = n2 - 1 denominator degrees of freedom. The F- distribution has a positive skew.
(i)In two way ANOVA, the data are classified according to more than one criterion.
Example: A tea company appoints four salesmen A, B, C and D and records their sales in three seasons - summer, winter and monsoon.