Question

1. Verbally describe how SSwithin is computed in one-way ANOVA, then represent that using graphs depicting the distribution of scores in the three groups.

2. Verbally describe how SSbetween is computed in one-way ANOVA, then represent that using graphs depicting the distribution of scores in the three groups

3. Verbally describe how SStotal is computed in a one-way ANOVA, then represent that using graphs depicting the distribution of scores in the three groups.

Answer 1

Total Variation : The total variation is calculated by,

Where,

Grand Mean

The grand mean of a set of samples is the total of all the data values divided by the total sample size.

The total variation (not variance) is comprised the sum of the squares of the differences of each mean with the grand mean.

There is the between group variation and the within group variation. The whole idea behind the analysis of variance is to compare the ratio of between group variance to within group variance. If the variance caused by the interaction between the samples is much larger when compared to the variance that appears within each group, then it is because the means aren't the same.

Between Group Variation

The variation due to the interaction between the samples is denoted SS(B) for Sum of Squares Between groups. If the sample means are close to each other (and therefore the Grand Mean) this will be small. There are k samples involved with one data value for each sample (the sample mean), so there are k-1 degrees of freedom.

The variance due to the interaction between the samples is denoted MS(B) for Mean Square Between groups. This is the between group variation divided by its degrees of freedom. It is also denoted by .

Within Group Variation

The variation due to differences within individual samples, denoted SS(W) for Sum of Squares Within groups. Each sample is considered independently, no interaction between samples is involved. The degrees of freedom is equal to the sum of the individual degrees of freedom for each sample. Since each sample has degrees of freedom equal to one less than their sample sizes, and there are k samples, the total degrees of freedom is k less than the total sample size: df = N - k.

The variance due to the differences within individual samples is denoted MS(W) for Mean Square Within groups. This is the within group variation divided by its degrees of freedom. It is also denoted by . It is the weighted average of the variances (weighted with the degrees of freedom).

Source	Degrees of freedom	SS	MS	F
Between	k-1	SS(B)	SS(B)/(k-1)	MS(B)/MS(W)
Within	N-k	SS(W)	SS(W)/(N-k)
Total	N-1	SS(B)+SS(W)