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ANOVA Formula

Last updated date: 24th Feb 2024
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Define ANOVA

The ANOVA formula is the abbreviation of the Analysis of the variance formula. Analysis of variance (ANOVA) is an analysis tool used in statistics that segregates an observed mean variability found inside a data set into two parts namely the systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Examiners apply the ANOVA test to define the influence that independent variables have on the dependent variable in a regression study.

In this article, we will study the ANOVA formula, ANOVA full form, and ANOVA statistics with solved ANOVA example problems. 

Anova Full Form

The ANOVA full form is the Analysis of variance formula, the ANOVA formula is a strong statistical technique and it is generally used to show the variation between two or more means or components through consequence tests. The ANOVA full form and by the way we define ANOVA it will help us to show a way to make multiple comparisons of several populations. 

The Anova formula is used by comparing two types of variation, the variation between the sample means, as well as the variation within each of the samples. The below-mentioned formula represents one-way Anova test statistics.

The ANOVA formula is given by:

⇒ F = \[\frac{MST}{MSE}\]…… (1) 


F - The ANOVA coefficient

MST - The mean sum of all the squares due to the treatment

MSE - The mean sum of squares due to error

Equation (1) is known as the ANOVA formula and the ANOVA full form is the analysis of the variance formula.

ANOVA Statistics

The ANOVA formula will be varied depending upon the consideration of variance. It implies that the ANOVA formula can be re-written for the different variance range i.e., for variance obtained within the data points, between the data points, etc.

Basically, the ANOVA test formula will allow us to practice a comparison of more than two groups simultaneously to determine whether a relationship exists between them. The outcome of the ANOVA statistics formula is known as the F statistic (also called the F-ratio or ANOVA statistics), and it allows us for the analysis of recurring groups of data points to determine the variance between samples and within samples.

If no true difference exists between the groups considered for testing (analysis of variance example), then it is known as the null hypothesis, the outcome of the ANOVA formula F-ratio statistic will always be nearly 1 or equal to 1. The arrangement of all possible values of the F statistic is the F-distribution. This is actually a group of distribution functions, with two characteristic numbers, called the numerator degrees of freedom and the denominator degrees of freedom.

The ANOVA formula and ANOVA method for the variance between the samples and within the samples is given by:

⇒ F = \[\frac{MST}{MSE}\]

⇒ F = \[\frac{\sum_{j=1}^{k}\sum_{j=1}^{l}(\bar{x}_{j}-x_{j})^{2}}{df_{w}}\]

⇒ F = \[\frac{\sum_{j=1}^{k}\sum_{j=1}^{l}(\bar{x}_{j}-x_{j})^{2}}{k-1}\]


x - The data points

\[\bar{x}\]\[_{j}\] - The mean of the data points

df\[_{w}\] - The degrees of freedom for the data points within the range

Anova Examples

1. Consider the following data to interpret the quantities of worms quarantined from the GI regions of four groups of muskrats in a trial of carbon tetrachloride as an anthelmintic. These four groups were the restraint (untreated) groups.

Expt 1

Expt 2

Expt 3

Expt 4






















Sol: To analyse the given data, we must first prepare the required calculations such as the Mean sum value, total sum value, etc. our main aim is to determine the i.e, variance considered between the data points and within the data points. Let us first tabulate those numbers for simplification.

Source of Variation

Sum Squares

Degrees of Freedom

Mean Square

Between the groups




Within the groups




Corrected total



Now, from the above-tabulated data, we calculate the ANOVA variance using the below formula 

(ANOVA Table Formulas):

⇒ F = \[\frac{MST}{MSE}\] …… (1) 


F - The ANOVA coefficient

MSB - The mean sum of all the squares due to the treatment

MSE - The mean sum of squares due to error

Substituting all the values and simplify:

⇒ F = 2.2711 P = 0.1195 

Here, P < 0.05%

Assuming there is no difference in the mean worm count of the four groups, this hypothesis holds. If we reject this null hypothesis, then we will have to carefully review the experimental conditions to ensure that all control groups are exposed to the same conditions.

FAQs on ANOVA Formula

1) How Do We Calculate One-way ANOVA?

Answer: To calculate the one way ANOVA formula we follow these steps mentioned below:

Step 1: Estimate the total group means and the overall mean. Initially, we will estimate the mean for all three groups along with the overall mean.

Step 2: Then Calculate the sum of squares

Step 3: After SSR calculate Sum of Squares of Error.

Step 4: Calculate Sum of squares of transitions

Step 5: Substitute in the ANOVA table.

Step 6: Interpret the results.

2) What is k in the ANOVA Formula?

Answer: K represents the total number of independent groups.