## Table of Contents:-

- What is analysis of variance?
- Use of Anova
- Characteristics of Analysis of Variance (ANOVA)
- Assumptions of Analysis of Variance (ANOVA)
- Applications of ANOVA
- Basic Principle of ANOVA
- ANOVA Techniques

## What is analysis of variance?

The Analysis of VarianceÂ is a statistical method thatÂ is similar to the regression analysis.Â It is abbreviated as ANOVA, developed by R.A. Fisher. ANOVA is widely used by researchers to investigate the underlying relationship between dependent variables and various independent variables.

It is widely used in the areas of business, industries, economics, psychology, sociology, and biology the researchers prefer the ANOVA when there are multiple cases of the sample. It is a statistical tool which is used to analyse the variations in the data and calculate the inequality among the population’s means.

Using a z-test or t-test, the researcher does the study to find the significance of the difference between the means of the two samples but the problem arises when he tries to measure the significance of the difference of more than two sample means simultaneously.

The statistical tool ANOVA removed this problem. By using ANOVA, the researcher can conclude, that the samples collected are from the population which have the same mean.

For example, it is useful when the researcher wants to examine the side effects of medicine on different age groups, the mileage of various motorbikes, the eating habits of various cultures, etc.

In these situations, the researchers do not want to examine all the combinations of the two or more than two populations simultaneously, because it will require a lot of research tests to come to any conclusion.

Moreover, it requires a researcher to invest in a lot of resources, including financial capital and time. In these kinds of situations, the researchers prefer to use ANOVA as it helps them to examine the differences in the sample means of various populations at the same time

## Use of Anova

ANOVA is used to check the level of homogeneity among the groups of samples from different populations. The use of ANOVA by researchers helps them to divide the total variation of the various samples into two groups.

- The first variation is that which occurs from random chance and
- The second variation is that which occurs due to any specific cause.

ANOVA helps the researcher, to split the variance for investigative purposes. With the use of ANOVA, the researchers can examine the ‘n’ number of factors influencing the dependent variables. In simpler words, Analysis of Variance (ANOVA) is a statistical technique which is used for exploring the variation of a dependent variable concerning independent variables.

In the ANOVA technique when the single factor is taken and the researcher analyses the difference amongst its various categories which have numerous possible values, then this ANOVA technique is known as one-way ANOVA When the analysis is done on two factors at the same time, then this ANOVA is known as two-way ANOVA.

## Characteristics of Analysis of Variance (ANOVA)

**1)** One of the most important characteristics of ANOVA is that it splits the variance into chance variance and cause variance.

**2)** ANOVA results are free from any type of constant bias and scaling errors.

**3)** It examines the variance of more than two groups of samples.

**4)** ANOVA is commonly used to analyze and compare multiple groups in experiments.Â

**5)** The ANOVA research tool is helpful in the study of analysis of variance.

**6)** It describes theÂ accurateÂ and complete characteristics ofÂ specificÂ populations.

## Assumptions of Analysis of Variance (ANOVA)

Analysis of variance is done based on some assumptions. These assumptions are given below:

**1)** Samples are randomly selected to ensure homogeneity and independence of each other.

**2)** Starts with the null hypothesis i.e., V_{1} = V_{2} = V_{3} = ….V_{n} (where, V_{1} ,V_{2} ,V_{3} ….V_{n} are variance).

**3)** Normally distributed universe used to take the sample for the study.

**4)** It is assumed that the critical values of the variance ratio, (F) as estimated by Suedecar for different degrees of freedom at different levels of significance, viz.; 5%, or 1%, etc., hold good in testing the significance of the calculated values of F that represents the variance ratio of the samples.

**5)** It is assumed that there is no significant difference amongst the variances of the various universes from which the samples have been drawn.

## Applications of ANOVA

**1)** If medical researchers or scientists are interested in studying the impact of medicine on different age groups of human beings, then ANOVA is very useful.

**2)** The statistical tool ANOVA can help the researcher to study whether the purchase intentions of a group of people differ significantly or not.

**3)** With the help of the ANOVA an organic scientist or researcher can elaborate on whether different varieties of fertilizers, seeds or soils differ significantly or not. This helps the researchers draft the policy according to the outcomes.

**4)** The top management of an organization may examine the performance of their sales managers working on different populations. Using ANOVA, management can decide whether the performance or the efficiency of the sales managers differs significantly or not.

## Basic Principle of ANOVA

The underlying principle of the ANOVA is to compare the differences in the different means of population, by studying the amount of the variations within the samples, relative to the variations between samples of the population.

In the case of the variations which occur inside a given population, generally, it is presumed by the researcher that this is because of the random effect in the population. In simpler words, because of the random effect within the population, the value of Xij (a random value) differs from the mean of that population,

It has been observed that there exist some influences on the (Xij), which cannot be explainable. On the other hand, the differences which occur amongÂ populations are due to some specific factor. This is the reason why the researchers assume using the ANOVA technique; that the samples have been selected from the normal population.

Also, each population has the same degree of variance. It is assumed by the researcher that there are a few variables which can’t be controlled but the rest of the variables are easily controllable. So, the researcher assumes that the absence of various factors may affect the results regarding the factor(s) to be studied. While estimating the ANOVA, the researcher needs the estimate the two population variances:

- Based on Sample Variance and
- Between the Sample Variance.

The values of the above two estimates of the population are compared with the F-test. The F-test is calculated as follows:

$$\mathrm F\;=\;\;\frac{\mathrm{Estimation}\;\mathrm{of}\;\mathrm{the}\;\mathrm{population}\;\mathrm{variance}\;\mathrm{based}\;\mathrm{on}\;\mathrm{the}\;\mathrm{sample}\;\mathrm{variance}}{\mathrm{Estimation}\;\mathrm{of}\;\mathrm{the}\;\mathrm{population}\;\mathrm{variance}\;\mathrm{based}\;\mathrm{on}\;\mathrm{within}\;\mathrm{sample}\;\mathrm{variance}}$$

The above-calculated value of the F-test is then compared with the limit of the F-test for a given degree of freedom. If the calculated value of the above equation is more than or equal to F-Limit, then there exists a significant change between the sample means.

## ANOVA Techniques

Following are the two ANOVA techniques:

- Variance Which Occurs Due to the One Variable
- Variance Which Occurs Due to Two Variables.

The above-stated techniques of the variances are discussed in detail which is as follows:

### 1) Variance which occurs due to the One Variable

In this case, the change or the variance that occurs in a variable is because of only a single variable. Or we can say that there is only one independent variable and we try to study the impact of that independent variable on the other dependent variable

For example, if the researcher wants to study the impact of one variable (e.g., anyone amongst types of seed or fertilizer or soil) on the other variable (i.e., amount of harvest of a crop), then this is known as variance due to the one variable or one-way ANOVA. Here, the type of fertilizer, soil and seeds are three independent variables. In this case, the researcher will study the impact of one independent variable (from seed, fertilizer or soil) on the dependent variable (harvesting of crop). This type of relationship can be given as:

### 2) Variance which occurs due to the Two Variables

In this case, the researcher wants to study the impact of multiple independent variables on a single dependent variable. Here, the impact of the changes or the variation in two or more independent variables is studied in different conditions and the values corresponding dependent variable are also observed.

For example, if a researcher wants to study the impact of two independent variables, say season and effort of the salesman, on the total sales of the corresponding salesman. This type of situation is the best example of the variation due to the impact of two variables. The above example can be best described with the help of the below-stated format.

Based on the above classification, there are two ANOVA techniques:

- Analysis of the variance for the one-way classification (One way ANOVA).
- Analysis of the variance for the two-way classification (Two way ANOVA).

Â