Anova Formula

ANOVA Test, or Analysis of Variance, is a statistical method used to test the differences between the means of two or more groups. Developed by Ronald Fisher in the early 20th century, ANOVA helps determine whether there are any statistically significant differences between the means of three or more independent groups.

The ANOVA analysis is a statistical relevance tool designed to evaluate whether or not the null hypothesis can be rejected while testing hypotheses. It is used to determine whether or not the means of three or more groups are equal.

The ANOVA test is used to look for heterogeneity within groups as well as variability across groupings. The F-test returns the ANOVA test statistic.

ANOVA Formula

ANOVA formula is made up of numerous parts. The best way to tackle an ANOVA test problem is to organize the formulae inside an ANOVA table. Below are the ANOVA formulae.

where,

• F = ANOVA Coefficient
• MSB = Mean of the total of squares between groupings
• MSW = Mean total of squares within groupings
• MSE = Mean sum of squares due to error
• SST = total Sum of squares
• p = Total number of populations
• n = The total number of samples in a population
• SSW = Sum of squares within the groups
• SSB = Sum of squares between the groups
• SSE = Sum of squares due to error
• s = Standard deviation of the samples
• N = Total number of observations

Examples of the use of the ANOVA Formula

• Assume it is necessary to assess whether consuming a specific type of tea will result in a mean weight decrease. Allow three groups to use three different varieties of tea: green tea, Earl Grey tea, and Jasmine tea. Thus, the ANOVA test (one way) will be utilized to examine if there was any mean weight decrease displayed by a certain group.
• Assume a poll was undertaken to see if there is a relationship between salary and gender and stress levels during job interviews. A two-way ANOVA will be utilized to carry out such a test.

ANOVA Table

An ANOVA (Analysis of Variance) test table is used to summarize the results of an ANOVA test, which is used to determine if there are any statistically significant differences between the means of three or more independent groups. Here's a general structure of an ANOVA table:

Types of ANOVA  Formula

One-Way ANOVA

This test is used to see if there is a variation in the mean values of three or more groups. Such a test is used where the data set has only one independent variable. If the test statistic exceeds the critical value, the null hypothesis is rejected, and the averages of at least two different groups are significant statistically.

Two-Way ANOVA

Two independent variables are used in the two-way ANOVA. As a result, it can be viewed as an extension of a one-way ANOVA in which only one variable influences the dependent variable. A two-way ANOVA test is used to determine the main effect of each independent variable and whether there is an interaction effect. Each factor is examined independently to determine the main effect, as in a one-way ANOVA. Furthermore, all components are analyzed at the same time to test the interaction impact.

Solved Examples on ANOVA Formula

Example 1: Three different kinds of food are tested on three groups of rats for 5 weeks. The objective is to check the difference in mean weight(in grams) of the rats per week. Apply one-way ANOVA using a 0.05 significance level to the following data:

Solution:

H₀: μ₁= μ₂=μ₃
H₁: The means are not equal

Since, X̄₁ = 5, X̄₂ = 9, X̄₃ = 10

Total mean = X̄ = 8

• SSB = 6(5 - 8)² + 6(9 - 8)² + 6(10 - 8)² = 84
• SSE = 68
• MSB = SSB/df₁ = 42
• MSE = SSE/df₂ = 4.53
• f = MSB/MSE = 42/4.53 = 9.33

Since f > F, the null hypothesis stands rejected.

Example 2: Calculate the ANOVA coefficient for the following data:

Solution:

Plant	n	x	s	s2
Hibiscus	5	12	2	4
Marigold	5	16	1	1
Rose	5	20	4	16

p = 3
n = 5
N = 15
x̄ = 16

SST = Σn(x−x̄)²

• SST= 5(12 − 16)² + 5(16 − 16)² + 11(20 − 16)² = 160
• MST = SST/p-1 = 160/3-1 = 80
• SSE = ∑ (n−1) = 4 (4 + 1) + 4(16) = 84
• MSE = 7
• F = MST/MSE = 80/7
• F = 11.429

Example 3: The following data show the number of worms quarantined from the GI areas of four groups of muskrats in a carbon tetrachloride anthelmintic study. Conduct a two-way ANOVA test.

Solution:

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square
Between the groups	62111.6	8	9078.067
Within the groups	98787.8	16	4567.89
Total	167771.4	24

Since F = MST / MSE
= 9.4062 / 3.66
F = 2.57

Example 4: Enlist the results in APA format after performing ANOVA on the following data set:

\begin{bmatrix}  \textbf{n} & \textbf{mean} & \textbf{sd} \\  30 & 50.26 & 10.45 \\  30 & 45.32 & 12.76 \\  30 & 53.67 & 11.47 \\ \end{bmatrix}

Solution:

• Variance of first set = (10.45)² = 109.2
• Variance of second set = (12.76)² = 162.82
• Variance of third set = (11.47)² = 131.56

MS_error = {109.2 + 162.82 + 131.56} / {3}
= 134.53

MS_between = (17.62)(30) = 528.75
F = MS_between  /  MS_error 
  = 528.75 / 134.53
F = 4.86

APA writeup: F(2, 87)=3.93, p >=0.01, η²=0.08.

Articles Related to the ANOVA  Formula:

• Variance and Standard Deviation
• How to Calculate Variance?
• Frequency Distribution

Annova for Programmers:

• One-Way ANOVA in Python
•  Two-Way ANOVA in Python
• ANOVA Testing in R