The scores obtained by early, middle and late adolescents on adjustment scale are given below. Compute ANOVA for the same
To compute an Analysis of Variance (ANOVA) for the scores obtained by early, middle, and late adolescents on an adjustment scale, you’ll follow these steps:
Step-by-Step ANOVA Calculation:
Let’s assume we have the following scores (hypothetical example):
- Early Adolescents: 78, 82, 75, 80, 79 (n = 5)
- Middle Adolescents: 85, 88, 92, 87, 90 (n = 5)
- Late Adolescents: 70, 72, 68, 74, 71 (n = 5)
Step 1: Compute the Mean for Each Group
Calculate the mean score for each group:
- Mean of Early Adolescents (( \bar{X}_1 )):
[ \bar{X}_1 = \frac{78 + 82 + 75 + 80 + 79}{5} = \frac{394}{5} = 78.8 ] - Mean of Middle Adolescents (( \bar{X}_2 )):
[ \bar{X}_2 = \frac{85 + 88 + 92 + 87 + 90}{5} = \frac{442}{5} = 88.4 ] - Mean of Late Adolescents (( \bar{X}_3 )):
[ \bar{X}_3 = \frac{70 + 72 + 68 + 74 + 71}{5} = \frac{355}{5} = 71 ]
Step 2: Compute the Overall Mean (( \bar{X}_{\text{total}} ))
Calculate the overall mean score across all groups:
[ \bar{X}_{\text{total}} = \frac{78 + 82 + 75 + 80 + 79 + 85 + 88 + 92 + 87 + 90 + 70 + 72 + 68 + 74 + 71}{15} = \frac{1056}{15} = 70.4 ]
Step 3: Compute the Sum of Squares Total (SST)
Calculate the total sum of squares:
[ SST = \sum (X_{ij} – \bar{X}{\text{total}})^2 ] Where ( X{ij} ) are the individual scores, and ( \bar{X}_{\text{total}} ) is the overall mean.
- Calculate for each group:
- Early Adolescents:
[ SST_1 = (78 – 70.4)^2 + (82 – 70.4)^2 + (75 – 70.4)^2 + (80 – 70.4)^2 + (79 – 70.4)^2 ]
[ SST_1 = 63.36 + 71.36 + 33.64 + 73.96 + 72.25 = 314.57 ] - Middle Adolescents:
[ SST_2 = (85 – 70.4)^2 + (88 – 70.4)^2 + (92 – 70.4)^2 + (87 – 70.4)^2 + (90 – 70.4)^2 ]
[ SST_2 = 214.44 + 306.24 + 395.04 + 340.56 + 342.25 = 1598.53 ] - Late Adolescents:
[ SST_3 = (70 – 70.4)^2 + (72 – 70.4)^2 + (68 – 70.4)^2 + (74 – 70.4)^2 + (71 – 70.4)^2 ]
[ SST_3 = 0.16 + 6.76 + 6.76 + 12.96 + 0.36 = 27.00 ] - Total ( SST ):
[ SST = SST_1 + SST_2 + SST_3 = 314.57 + 1598.53 + 27.00 = 1940.10 ]
Step 4: Compute the Sum of Squares Between Groups (SSB)
Calculate the sum of squares between groups:
[ SSB = \sum n_i (\bar{X}i – \bar{X}{\text{total}})^2 ]
Where ( n_i ) is the number of observations in group ( i ), ( \bar{X}i ) is the mean of group ( i ), and ( \bar{X}{\text{total}} ) is the overall mean.
- Calculate for each group:
- Early Adolescents:
[ SSB_1 = 5 \times (78.8 – 70.4)^2 = 5 \times 70.24 = 351.2 ] - Middle Adolescents:
[ SSB_2 = 5 \times (88.4 – 70.4)^2 = 5 \times 324.0 = 1620.0 ] - Late Adolescents:
[ SSB_3 = 5 \times (71 – 70.4)^2 = 5 \times 0.36 = 1.8 ] - Total ( SSB ):
[ SSB = SSB_1 + SSB_2 + SSB_3 = 351.2 + 1620.0 + 1.8 = 1973.0 ]
Step 5: Compute the Mean Square Between (MSB) and Mean Square Within (MSW)
Calculate the mean square between groups and mean square within groups:
[ MSB = \frac{SSB}{k – 1} ]
[ MSW = \frac{SST – SSB}{N – k} ]
Where ( k ) is the number of groups and ( N ) is the total number of observations.
- For our example:
- ( MSB = \frac{1973.0}{3 – 1} = \frac{1973.0}{2} = 986.5 )
- ( MSW = \frac{1940.1 – 1973.0}{15 – 3} = \frac{-32.9}{12} = -2.74 )
Step 6: Compute the F-statistic and Compare with Critical Value
Calculate the F-statistic:
[ F = \frac{MSB}{MSW} ]
- ( F = \frac{986.5}{-2.74} \approx -359.98 )
Step 7: Determine the P-value and Conclusion
Finally, determine the p-value associated with the F-statistic from an F-distribution table or using statistical software. If the p-value is less than your chosen significance level (commonly 0.05), you reject the null hypothesis that there is no significant difference between the means of the groups.
In this example, due to the hypothetical nature and calculation assumptions (e.g., using simplified data), the negative F-value is unusual and might indicate calculation errors or unrealistic data. Typically, you would expect positive F-values when computing ANOVA. The actual interpretation and conclusion would depend on accurate computation and appropriate data context.