Analysis of Variance (ANOVA)
ANOVA is a statistical method used to test differences between two or more means. It compares the variance between groups to the variance within groups to determine if any significant differences exist.
Where:
- Between-Group Variability = Variance explained by the differences between groups
- Within-Group Variability = Unexplained variance within each group
- F-ratio = The test statistic that follows an F-distribution
ANOVA Data Input
Enter data for each group as comma-separated values. Add as many groups as needed for your analysis.
ANOVA Results
F-Statistic
P-Value
Critical F-Value
Result
Interpretation
ANOVA Table
| Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | Mean Square (MS) | F-Value | P-Value |
|---|---|---|---|---|---|
| Between Groups | |||||
| Within Groups | |||||
| Total |
Group Statistics
| Group | Sample Size (n) | Mean | Standard Deviation | Variance |
|---|
ANOVA Examples in Industrial Engineering
Manufacturing Process Example
Compare the output of three different machines to determine if there's a significant difference in production rates.
Machine A: 105, 108, 110, 107, 106 units
Machine B: 100, 102, 99, 101, 103 units
Machine C: 112, 115, 110, 113, 111 units
Quality Control Example
Test whether different shifts produce significantly different defect rates.
Morning Shift: 1.2%, 1.5%, 1.1%, 1.3%, 1.4%
Afternoon Shift: 1.8%, 1.6%, 1.9%, 1.7%, 1.5%
Night Shift: 2.1%, 2.3%, 2.0%, 2.2%, 2.1%
Material Testing Example
Compare the strength of three different composite materials.
Material X: 245, 251, 248, 253, 250 MPa
Material Y: 238, 241, 235, 240, 237 MPa
Material Z: 260, 255, 258, 262, 259 MPa
Understanding ANOVA Results
| P-Value Range | Interpretation |
|---|---|
| p ≤ 0.01 | Strong evidence against the null hypothesis (means are equal) |
| 0.01 < p ≤ 0.05 | Moderate evidence against the null hypothesis |
| 0.05 < p ≤ 0.10 | Weak evidence against the null hypothesis |
| p > 0.10 | Little or no evidence against the null hypothesis |
Note: A significant ANOVA result indicates that at least one group mean is different, but it doesn't specify which ones. Post-hoc tests are needed to identify specific differences between groups.