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Making the right decisions when faced with various solutions that all seem viable can be a challenging undertaking. It requires a thorough analysis to finally choose and back up your decision statistically. This is where analysis of variance (ANOVA) comes in. This statistical technique, especially in coursework, entails comparing various samples based on their means. It aims at determining whether these means significantly differ, or not and then arriving at a conclusion. To fully understand what the concept of analysis of variance is all about, the following terminologies are essential:

**Grand mean**

The mean represents an average of all the samples present. In ANOVA, the sample means and grand mean are used. The grand mean is the average of the sample means or the average of all combined observations. It is denoted by µ while the sample means are denoted by µ1, µ2, µ3 and so forth

**Hypothesis**

A hypothesis refers to a testable statement regarding a certain experiment. We have a null hypothesis and an alternate hypothesis. The null hypothesis is true when the sample means are equal or have no significant differences. The alternate hypothesis is valid when at least one sample mean has a considerable difference from the rest.

Null hypothesis Ho: µ1= µ2=…= µn

Alternate hypothesis H1: µ1≠ µn

Between-group variability

This refers to variations between distributions of individual groups with a difference in each group’s values.

**Within-group variability**

These are variations caused by differences within individual groups (or levels) due to differences in values of distinct groups.

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## Types of ANOVA

Analysis of variance can be one way, two way or multivariate depending on the number of samples under consideration.

### One-way ANOVA

One-way ANOVA uses categorical variables for the independent variable and a continuous variable for the dependent variable. These continuous variables are divided into groups whose mean differences are calculated to establish whether they are significantly different. For instance, one can seek to establish whether the different vaccines used to treat a certain similar ailment have statistically significant differences in means. After conducting the one-way ANOVA, if the p-value is less than the significant value (mostly 0.05), reject the null hypothesis. Another criterion of rejecting is if the F-value is greater than the F-critical value for the alpha level selected.

#### Limitations of one-way ANOVA

While one way ANOVA establishes that some groups are different from the others, it does not specify which ones. This calls for a post hoc test to get the specific groups that have significantly different means.

#### Steps to perform one-way ANOVA in excel

If you do not have the data analysis add-in already, click on excel file, options, and then add-ins. Click on the “Go” button then select data analysis and it will be added to your menu.

- Input your data in rows and columns in Excel. In the case of vaccines, enter the results from the different vaccines being tested.
- Click
*data analysis*on the data tab - Click
*ANOVA*: single factor then click ok - Select the
*input range*containing the values you want to analyze - Select
*grouped by*columns - Check the
*Labels in first row*box for easier interpretation of your results - An alpha value of 0.05 will already have been inputted
- Click
*new worksheet ply*button for the output range - Click
*ok*

### Two-way ANOVA

One-way ANOVA considers data within the same categories. What if these categories were to be extended? How would one establish whether there exists a difference in the sample means or not? The two-way ANOVA incorporates values within different categories; for instance age of patients in the vaccine experiment. This means there are factors to consider in the treatment being studied, resulting in two independent variables. Organize your data depending on the two factors in consideration. You are now resting two null hypotheses and a third hypothesis can exist as the alternative hypothesis, refuting the effect of the two factors. Reject the null hypothesis for a particular effect if its F value is greater than the F-critical value. Also, if the p-value is smaller than the alpha value you can cite that there is enough evidence to reject the null hypothesis.

#### Steps to perform two-way ANOVA in excel

- Click on
*data analysis*on the data tab - Click
*ANOVA*: two factor with replication and the two-way ANOVA window opens - Select the input range in which your data is contained, including the headers and group names
- In
*rows per sample*box, input the number of individuals you have - Select
*new worksheet ply*in output options - If not already specified, most tests use an alpha value of 0.05
- Click Ok to run the two-way ANOVA and check the results to determine whether you will reject the null hypothesis or not.

### MANOVA

Multivariate analysis of variance establishes whether one independent variable affects multiple dependent variables. One factor can have multiple levels that will also affect the dependent variables. It will have a null hypothesis: Ho: µ1= µ2=…= µn and an alternate hypothesis H1: µ1≠ µn.

#### Steps to perform MANOVA in excel

- If you do not have ‘RealStats’ add-in, download it
- Press “control + m” to open RealStats window
- Select
*Analysis of variance* - Click on
*MANOVA single factor* - Select the input range depending on the data in your worksheet, including the headers and group names
- Select
*Significance analysis*,*Group Means*and*Multiple ANOVA* - Select the preferred output range and alpha value (0.05 might be already selected in most instances)
- Click ok and read the results. Reject the null hypothesis if; F-value is greater than F-critical or if the p-value is smaller than the chosen alpha value.

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