Anova Formula

ANOVA Test (Analysis of Variance) is used to compare the means of different groups using various estimate methodologies. ANOVA is an abbreviation for the analysis of variance. 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. Whenever there are more than two or more independent groups, the ANOVA test is used. 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.

Source of Variation	Sum of Squares	Degree of Freedom	Mean Squares	F Value
Between Groups	SSB = Σnj(X̄j– X̄)2	df1 = k – 1	MSB = SSB / (k – 1)	f = MSB / MSE or, F = MST/MSE
Error	SSE = Σnj(X̄- X̄j)2	df2 = N – k	MSE = SSE / (N – k)
Total	SST = SSB + SSE	df3 = N – 1

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 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.

Types of ANOVA  Formula

1. 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.
2. 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.

ANOVA Formula Example

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:

Food I	Food II	Food III
8	4	11
12	5	8
19	4	7
8	6	13
6	9	7
11	7	9

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:

Plant	Number	Average span	s
Hibiscus	5	12	2
Marigold	5	16	1
Rose	5	20	4

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.

I	II	III	IV
338	412	124	389
324	387	353	432
268	400	469	255
147	233	222	133
309	212	111	265

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:

[Tex]\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} [/Tex]

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.

Related Resources:

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

ANOVA  Formula – FAQs

How does one set the hypothesis for ANOVA?

The equality of the means of distinct groups must be tested in an ANOVA test. As a result, the hypothesis is as follows:

H₀ = 1 = 2 = 3 =… = k = Null Hypothesis
H₁ Alternative Hypothesis: The means are not equal.

How do you calculate the ANOVA?

ANOVA compares group means’ differences. Calculate Grand Mean, Between-Group Variability (SSB), and Within-Group Variability (SSW). Determine significance through variance comparison.

What is meant by ANOVA statistic?

The sample mean of the jth treatment of a grouping or a mass data sample is called the ANOVA statistic. It is denoted by the alphabet f. 

What is the p value in ANOVA?

In ANOVA, a shared P-value is initially obtained. A significant P-value in the ANOVA test suggests statistical significance in at least one pair’s mean difference. Multiple comparisons are then employed to identify these significant pair(s).

What do you mean by one-way ANOVA?

One-way ANOVA is a form of ANOVA test that is used when just one independent variable is present. It compares the means of the various test groups. A test of this type can only provide information on the statistical significance of the means; it cannot establish which groups have different means.

Comment on the accuracy of the ANOVA test.

Since it is more versatile and requires fewer observations, ANOVA analysis is sometimes thought to be more accurate than t-testing. It is also more suited to employ in more sophisticated studies than those that can be evaluated by testing.