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  One-Way Analysis of Variance

One-Way ANOVA:

To test the equality of three or more means by using variances at one time is a one way analysis test.

Assumptions:

i) The populations from which the samples were obtained must be approximately or normally distributed.
ii) The samples should be independent.
iii) The variances of the populations should be equal.

Hypotheses:

In null hypothesis, all population means are equal; and the alternative hypothesis is that at least one means is different.
Lower case letters apply to the individual samples and capital letters apply to the whole set collectively. That is, n is one of numerous sample sizes; however N is the net sample size.

Grand Mean:

Grand mean of the set of samples is the total of all data values divided by the net sample size. This requires that you all have the sample data available to you that is generally the case, however not always. This turns out that all is essential to determine the performance of a one-way analysis of variance is the number of samples, sample means, sample variances and the sample sizes.

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The other way to determine the grand mean is to find out the weighted average of sample means. The weight applied is a sample size.

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Total Variation:

It is the total variation (not variance) is comprised of the sum of squares of the differences of each and every mean with grand mean.
 
There is a group variation between and the in group variation. The entire idea behind the analysis of variance is to compare the ratio of between group variance to the within group variance. When the variance caused by interaction among the samples is much bigger if compared to the variance which appears in each group, then it is since the means are not similar.

Between Group Variation:

The variation due to interaction among the samples is symbolized SS(B) for the Sum of Squares among groups. When the sample means are close to one other (and thus the Grand Mean) this will be minimum. There are k samples comprised with one data value for each and every sample (that is, the sample mean), therefore there are (k-1) degree of freedom.

The variance due to interaction among the samples is symbolized by MS(B) for Mean Square among groups. This is between group variations divided by its degree of freedom. It is as well symbolized by sb2.

Within Group Variation:

It is the variation due to differences in individual samples, symbolized by SS(W) for the Sum of Squares in groups. Each and every sample is considered independently, no interaction among samples is included. The degree of freedom is equivalent to the sum of individual degree of freedom for each and every sample. As each sample has degree of freedom equivalent to one less than their sample sizes, and there are k samples, then the total degrees of freedom is k less than the net sample size: df = N - k.

The variance due to differences in individual samples is symbolized by MS(W) for Mean Square in groups. This is in group variation divided by its degree of freedom. It is as well symbolized by sw2. This is the weighted average of variances (that is, weighted with the degrees of freedom).

F test statistic:

Remember that an F variable is the ratio of two independent chi-square variables divided by their corresponding degree of freedom. As well remember that F test statistic is the ratio of two sample variances, well, it turns out that is precisely what we have here. The F test statistic is found out by dividing the between group variance by the in group variance. The degree of freedom for numerator is the degree of freedom for between group (k-1) and the degree of freedom for denominator is the degree of freedom for within group (N-k).

Summary Table:

All this sounds similar to a lot to memorize, and it is. Though, there is a table that makes things really good.

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Note that each of the Mean Square is simply the Sum of Squares divided by its degree of freedom and the F value is the ratio of mean squares. Do not place the biggest variance in the numerator, always divide between variance by the within variance. When the between variance is smaller than within variance, then the means are actually close to one other and you will fail to refuse the claim that they are all equivalent. The degree of freedom of F-test is in similar order that they appear in the table.

Decision Rule:

The decision will be to refuse the null hypothesis when the test statistic from the table is bigger than the F critical value with (k-1) numerator and (N-k) denominator degree of freedom.

When the decision is to refuse the null, then at least one of means is distinct. Though, the ANOVA doesn’t tell you where the differences lie. For this, you require other test, either the Scheffe' or the Tukey test.

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