Anova Calculator

ANOVA calculator is used to calculate the difference between two or more means or components through significance tests.

What is ANOVA?

ANOVA stands for Analysis of Variance, a statistical technique used to analyze and compare the means of two or more data groups.

The ANOVA test measures whether there is a significant difference between the means of the groups being compared or if the differences are simply due to chance.

Formulas for ANOVA:

All the following formulas are used while calculating the analysis of variance (ANOVA).

Formula for sum of squares between the group:

SSB = ∑_i=1^kn_i(X̄_i- X̄)²

∑ n_i: the sum of the number of observations in each group
X̄_i: the mean of the ith group
X̄: the overall mean of all the groups

Formula for sum of squares within the group:

SSW = ∑_i=1^K(n_i– 1) S_i²

X̄_i: the mean of the ith group
X̄: the overall mean of all the groups

Formula for the total sum of squares:

SST = SSB + SSW

SST represents the total sum of squares

Formula for mean square between groups:

MSB = SSB / (k – 1)

SSB represents the sum of squares between the group
K – 1 represents the degree of freedom of the group

Formula for mean square within the group:

MSW= SSW / (n – k)

SSW represents the sum of squares within the group
n represents the total sample size
k represents the total number of groups

Formula for test Statistics F:

F = MSB/MSW

MSB represents the mean square between groups
MSW represents the mean square within a group

Applications of ANOVA:

The analysis of variance helps us a lot in various fields. Some applications of ANOVA are as follows:

Medical research
Social sciences
Manufacturing
Agriculture
Business

Example section:

In this section, we’ll learn how the ANOVA table helps us solve our daily problems.

Example 1:

The times required by three workers to perform an assembly-line task were recorded on five randomly selected occasions. Here are the times, to the nearest minute.

Jim	Kane	John
5	0	6
1	1	9
4	4	8
2	6	7
8	3	5

Construct the one-way ANOVA table for the data. Compute SSB, SSW, MSB, MSW, and Static test F using the defining formulas.

Solution:

Step 1: Sum of squares between group

SSB = ∑_i=1^kn_i(X̄_i- X̄)²

SSB = 5 × (4 - 4.6)²+ 5× (2.8 - 4.6)²+ 5 × (7 - 4.6)²

SSB = 46.8

Step 2: Sum of squares within the group

SSW = ∑_i=1^K(n_i-1) Si²

SSW = (5 - 1) × (2.7386)²+ (5 - 1) × (2.3875)²+ (5 - 1) × (1.5811)²

SSW = 62.8

Step 3: Total sum of squares

SST = SSB + SSW

SST = 46.8 + 62.8

SST = 109.6

Step 4: Mean square between group

MSB = SSB / (k - 1)

MSB = 46.8 / (3 - 1)

MSB = 46.8 / 2

MSB = 23.4

Step 5: Mean square within the group

MSW = SSW / (n - k)

MSW = 62.8 / (15 - 3)

MSW = 5.233

Step 6: Test statistics F

F = MSB / MSW

F = 23.4 / 5.233

F = 4.472