Consider the following ANOVA table. Source of variation Degrees of freedom The sum of Squares (SS) Mean SS F Ratio Treatments a b c 5 Error 12 d 20 Total 15 540
The values of a, b, c and d are, respectively:
3, 300, 100 and 240
Understanding the ANOVA Table Structure
An Analysis of Variance (ANOVA) table summarizes the results of an ANOVA test. It breaks down the total variation observed in data into different sources. The standard components of an ANOVA table typically include:
- Source of Variation: Identifies where the variation is coming from (e.g., Treatments, Error, Total).
- Degrees of Freedom (DF): Represents the number of independent pieces of information available to estimate variation.
- Sum of Squares (SS): Measures the total variation within each source.
- Mean Sum of Squares (Mean SS or MS): Calculated by dividing the Sum of Squares by the Degrees of Freedom ($$MS = \frac{SS}{DF}$$). It represents the variance for each source.
- F Ratio: The ratio of the Mean SS for the source of interest (usually treatments or groups) to the Mean SS for the error ($$F = \frac{MS_{Treatment}}{MS_{Error}}$$). It is used to test the hypothesis about the equality of means.
There are specific relationships between the values in the table:
- $$DF_{Total} = DF_{Treatments} + DF_{Error}$$
- $$SS_{Total} = SS_{Treatments} + SS_{Error}$$
Analyzing the Given ANOVA Table
We are given the following incomplete ANOVA table:
| Source of Variation | Degrees of Freedom (DF) | Sum of Squares (SS) | Mean SS (MS) | F Ratio |
|---|---|---|---|---|
| Treatments | a | b | c | 5 |
| Error | 12 | d | 20 | |
| Total | 15 | 540 |
We need to find the values of a, b, c, and d.
Calculating the Unknown Values (a, b, c, d)
Step 1: Calculate 'a' (Treatments DF)
Using the relationship between degrees of freedom:
$$DF_{Total} = DF_{Treatments} + DF_{Error}$$ $$15 = a + 12$$ $$a = 15 - 12$$ $$a = 3$$So, the value of 'a' is 3.
Step 2: Calculate 'd' (Error SS)
Using the relationship between Mean SS, SS, and DF for the Error row:
$$MS_{Error} = \frac{SS_{Error}}{DF_{Error}}$$ $$20 = \frac{d}{12}$$ $$d = 20 \times 12$$ $$d = 240$$So, the value of 'd' is 240.
Step 3: Calculate 'b' (Treatments SS)
Using the relationship between sum of squares:
$$SS_{Total} = SS_{Treatments} + SS_{Error}$$ $$540 = b + d$$We know d = 240, so:
$$540 = b + 240$$ $$b = 540 - 240$$ $$b = 300$$So, the value of 'b' is 300.
Step 4: Calculate 'c' (Treatments Mean SS)
Using the relationship between Mean SS, SS, and DF for the Treatments row:
$$MS_{Treatments} = \frac{SS_{Treatments}}{DF_{Treatments}}$$ $$c = \frac{b}{a}$$We know a = 3 and b = 300, so:
$$c = \frac{300}{3}$$ $$c = 100$$So, the value of 'c' is 100.
Step 5: Verify the F Ratio
The F Ratio is calculated as:
$$F = \frac{MS_{Treatments}}{MS_{Error}}$$ $$F = \frac{c}{20}$$We know c = 100, so:
$$F = \frac{100}{20}$$ $$F = 5$$This calculated F ratio matches the value given in the table, confirming our calculations are correct.
Summary of Calculated Values
The calculated values are:
- a = 3
- b = 300
- c = 100
- d = 240
Thus, the values of a, b, c, and d are 3, 300, 100, and 240, respectively.
Revision Table: ANOVA Calculations
| Component | Formula | Example Calculation |
|---|---|---|
| Mean SS | $$MS = \frac{SS}{DF}$$ | $$MS_{Error} = \frac{d}{12} = 20 \implies d = 240$$ |
| DF (Total) | $$DF_{Total} = DF_{Treatments} + DF_{Error}$$ | $$15 = a + 12 \implies a = 3$$ |
| SS (Total) | $$SS_{Total} = SS_{Treatments} + SS_{Error}$$ | $$540 = b + d \implies 540 = b + 240 \implies b = 300$$ |
| F Ratio | $$F = \frac{MS_{Treatments}}{MS_{Error}}$$ | $$5 = \frac{c}{20} \implies c = 100$$ (Or $$c = \frac{b}{a} = \frac{300}{3} = 100$$) |
Additional Information: ANOVA Test
ANOVA (Analysis of Variance) is a statistical test used to compare the means of three or more groups to see if at least one group mean is statistically different from the others. It is based on partitioning the total variance in the data into different components attributable to different sources of variation. The F-statistic is the test statistic for ANOVA. A large F-statistic indicates that the variation between group means is large relative to the variation within groups, suggesting that at least one group mean is different. The F-statistic is then compared to a critical F-value from the F-distribution or used to calculate a p-value to determine statistical significance.
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