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}^{k }n_{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_{i}^{2}
- 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}^{k }n_{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) Si^{2}
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