Analysis of Variance (ANOVA) is a statistical method used to compare means among three or more groups to determine if at least one group mean is significantly different from the others. It helps in understanding whether the variations in data are due to actual differences between groups or just random noise.

1. Null Hypothesis (H0): Assumes that all group means are equal.
2. Alternative Hypothesis (H1): Assumes that at least one group mean is different.
3. F-Statistic: A ratio used to determine if the variances between groups are significantly different.
4. P-Value: The probability that the observed data would occur if the null hypothesis were true.

1. One-Way ANOVA: Compares means across one factor with multiple levels.
2. Two-Way ANOVA: Compares means across two factors, which can also include interaction effects between factors.
3. Repeated Measures ANOVA: Used when the same subjects are used for each treatment (i.e., repeated measurements).

Suppose we have three different teaching methods and we want to determine if there is a significant difference in the test scores of students taught by these methods.

1. State the Hypotheses:

   • \( H_0 \): μ1 = μ2 = μ3 (All group means are equal)
   • \( H_1 \): At least one group mean is different

2. Calculate the Group Means:

   • Compute the mean for each group.

3. Calculate the Overall Mean:

   • Compute the mean of all data points combined.

4. Calculate the Sum of Squares:

   • Between-Group Sum of Squares (SSB): Measures the variation due to the interaction between the groups.
   • Within-Group Sum of Squares (SSW): Measures the variation within each group.

5. Calculate the Mean Squares:

   • Mean Square Between (MSB): SSB divided by the degrees of freedom between groups.
   • Mean Square Within (MSW): SSW divided by the degrees of freedom within groups.

6. Calculate the F-Statistic:

   • \( F = \frac{MSB}{MSW} \)

7. Determine the P-Value:

   • Compare the F-statistic to the F-distribution to find the p-value.

8. Make a Decision:

   • If the p-value is less than the significance level (e.g., 0.05), reject the null hypothesis.

Given the following data:

Step-by-Step Calculation

1. Calculate Group Means:

   • Mean of Group A: \( \frac{85 + 90 + 88}{3} = 87.67 \)
   • Mean of Group B: \( \frac{78 + 82 + 80}{3} = 80 \)
   • Mean of Group C: \( \frac{92 + 94 + 91}{3} = 92.33 \)

2. Calculate Overall Mean:

   • Overall Mean: \( \frac{85 + 90 + 88 + 78 + 82 + 80 + 92 + 94 + 91}{9} = 85.67 \)

3. Calculate SSB and SSW:

   • SSB: \( 3 \times [(87.67 - 85.67)^2 + (80 - 85.67)^2 + (92.33 - 85.67)^2] = 3 \times [4 + 32.11 + 44.44] = 240.33 \)
   • SSW: \( (85 - 87.67)^2 + (90 - 87.67)^2 + (88 - 87.67)^2 + (78 - 80)^2 + (82 - 80)^2 + (80 - 80)^2 + (92 - 92.33)^2 + (94 - 92.33)^2 + (91 - 92.33)^2 = 4.67 + 5.44 + 0.11 + 4 + 4 + 0 + 0.11 + 2.78 + 1.78 = 22.89 \)

4. Calculate MSB and MSW:

   • MSB: \( \frac{240.33}{2} = 120.17 \)
   • MSW: \( \frac{22.89}{6} = 3.82 \)

5. Calculate F-Statistic:

   • \( F = \frac{120.17}{3.82} = 31.45 \)

6. Determine P-Value:

   • Using an F-distribution table or software, find the p-value corresponding to the F-statistic and degrees of freedom.

7. Decision:

   • If p-value < 0.05, reject \( H_0 \). In this case, the p-value is very small, so we reject \( H_0 \) and conclude that there is a significant difference in the means of the groups.

Given the following data, perform a one-way ANOVA to determine if there is a significant difference in the average weights of three different diets.

1. State the hypotheses.
2. Calculate the group means.
3. Calculate the overall mean.
4. Calculate SSB and SSW.
5. Calculate MSB and MSW.
6. Calculate the F-statistic.
7. Determine the p-value.
8. Make a decision.

1. Hypotheses:

   • \( H_0 \): μ1 = μ2 = μ3
   • \( H_1 \): At least one group mean is different

2. Group Means:

   • Mean of Diet 1: \( \frac{70 + 72 + 68}{3} = 70 \)
   • Mean of Diet 2: \( \frac{65 + 67 + 66}{3} = 66 \)
   • Mean of Diet 3: \( \frac{75 + 78 + 76}{3} = 76.33 \)

3. Overall Mean:

   • Overall Mean: \( \frac{70 + 72 + 68 + 65 + 67 + 66 + 75 + 78 + 76}{9} = 70.33 \)

4. SSB and SSW:

   • SSB: \( 3 \times [(70 - 70.33)^2 + (66 - 70.33)^2 + (76.33 - 70.33)^2] = 3 \times [0.11 + 18.78 + 35.78] = 162.21 \)
   • SSW: \( (70 - 70)^2 + (72 - 70)^2 + (68 - 70)^2 + (65 - 66)^2 + (67 - 66)^2 + (66 - 66)^2 + (75 - 76.33)^2 + (78 - 76.33)^2 + (76 - 76.33)^2 = 0 + 4 + 4 + 1 + 1 + 0 + 1.78 + 2.78 + 0.11 = 14.67 \)

5. MSB and MSW:

   • MSB: \( \frac{162.21}{2} = 81.11 \)
   • MSW: \( \frac{14.67}{6} = 2.45 \)

6. F-Statistic:

   • \( F = \frac{81.11}{2.45} = 33.11 \)

7. P-Value:

   • Using an F-distribution table or software, find the p-value corresponding to the F-statistic and degrees of freedom.

8. Decision:

   • If p-value < 0.05, reject \( H_0 \). In this case, the p-value is very small, so we reject \( H_0 \) and conclude that there is a significant difference in the means of the diets.

In this section, we have learned about the Analysis of Variance (ANOVA), its key concepts, and how to perform a one-way ANOVA. We also worked through a practical example and an exercise to reinforce the concepts. ANOVA is a powerful tool for comparing multiple group means and determining if there are significant differences among them.