## What is the Analysis of Variance?

Analysis of variance (ANOVA) based on a given claim compares three or more population means. A Z- tests and T-test have been widely used to analyze a single population means. A comparison of two population means is used to compare two population means. On some occasions, we are required to analyze by comparing means of more than two groups. In such cases, Analysis of Variance is applied. Analysis of Variance (ANOVA) is an analytical tool that helps compare variances across the population means of more than two groups.

The analysis of variance divides the population observed variability into within-group variance and between-group variances. Within-group variances have an impact on the given data set while the between-group variance does not have an impact on the given data set. The null hypothesis being tested isμ_1=μ_2=⋯=μ_kfor population involving K groups. F-statistics is used to conclude the difference in the means of the key groups. A significant difference between the groups will imply the difference between the group means. The conclusion will be to reject the null hypothesis. The F-ratio calculated in this situation will be greater than the F-critical value. In the case where the F critical value is greater than the F ratio calculated, we will lack evidence to conclude that there is no difference in group means and there will be no significant difference between groups.

## Definition of various analyses of variance terminology

While defining the terminologies used during the analysis of variance, we are going to consider an example below. A researcher wants to know whether there is a significant difference in average algebra scores of students from 3 universities. The major terms used during the analysis of various tests include:

- Determinant variable These are variables whose outcomes are influenced by the change of other variables. They are also known as outcome variables. From the example, the score of students is the outcome variable. These scores can be affected by the professors’ years of experience, availability of sufficient revision materials, and age how was the professor spent with the students Hence the score is the dependent variable.
- Independent variables Independent variables, also known as predictor variables, are the variables that will remain constant despite the change of other variables. When working on an analysis to record the scores of students in an algebra class; Gender, age, professor experience, and hours spent by the professor will not change hands we refer to them as independent variables.
- A null hypothesis Analyzing variance, a null hypothesis assumes that we do not have a notable statistical significance between the population mean. This is presented symbolically asH_0= μ_1=μ_2=⋯=μ_kWhen comparing means of K groups
- An alternative hypothesis An alternative hypothesis assumes that at least one of the groups has a different means from other groups. The symbolic representation of the null hypothesis is thatH_0=At least one means is different.
- F ratio This is a ratio of two average square values. It is calculated as a quotient of the mean sum of squares due to treatment and the mean sum of squares due to error. Ratios closer to 1 indicate a difference in the mean of different groups.
- Factor The predictor variables are known as factors because they influence the dependent variables.
- Levels Various values of the outcome variables are used in the experiment. A one-way ANOVA has only one level while after way ANOVA has more than one level.
- Fixed factor models When a researcher decides to use a fixed number of discrete levels in the analysis of variance, we see the model is a fixed factor. For example, the researcher might only decide to use three factors influencing scores (has taught, gender, and professors’ years of experience).
- Random factor model Random values are drawn from all possible predictor variable

## How has SPSS eased the Analysis of variance?

In SPSS, the analysis of variance can be categorized into two major groups.

- One-way analysis of variance (ANOVA)
- Two-way analysis of variance (ANOVA)

### 1. One-way ANOVA.

In one-way analysis of variance, the researcher is interested to investigate the mean difference between two or more groups based on a single fact. In SPSS running a descriptive statistic will give the sample size for each group, mean, and variance. A simple bar chart can be used to view the meaning of each group. To finally analyze the data, we use the command. Analyze Compare means way ANOVA. The analog box will appear. In the analog box, we specify the dependent variable and the factor variables. Click OK to run the commands.

### 2. Two-way analysis of variance (ANOVA)

Unlike the one-way analysis of variance, a two-way ANOVA investigates the mean difference of groups following the levels of categorical groups. In SPSS the procedure of two-way ANOVA is Analyze General Linear ModelUnivariate. This is specifically for one dependent variable and any number of independent variables. A dialog box will appear where we will specify a dependent variable or fixed factor (as) and random factors. Options level can be used to work out post hoc tests by finding the Turkeys Honestly significant difference, Waller-Duncan, LSD, Bonferroni, and S-N-K.

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