To determine whether there is a significant difference between the mean ages of marriage across different educational groups, one important step is to calculate the between-group sum of squares (SS_between). This statistic helps us understand the variation due to the differences between the group means relative to the overall mean.
Here's the detailed, step-by-step solution to compute the between-group sum of squares using the given data:
1. Identify the elements:
- Degrees: ['Less than high school', 'High school', 'Junior college', 'Bachelor\'s degree', 'Graduate degree']
- N: [195, 590, 86, 185, 104]
- Means: [22.47, 22.87, 23.71, 24.58, 25.10]
- Overall mean (mean of all groups combined): 23.33
2. Understand the formula:
The formula for the between-group sum of squares is:
[tex]\[
SS_{\text{between}} = \sum_{i} N_i (\overline{X}_i - \overline{X}_{\text{overall}}) ^ 2
\][/tex]
Where:
- [tex]\( N_i \)[/tex] is the number of observations in group [tex]\( i \)[/tex]
- [tex]\( \overline{X}_i \)[/tex] is the mean of group [tex]\( i \)[/tex]
- [tex]\( \overline{X}_{\text{overall}} \)[/tex] is the overall mean
3. Plug the values into the formula:
Calculate the contribution to [tex]\( SS_{\text{between}} \)[/tex] for each degree of education.
[tex]\[
\begin{align*}
\text{For 'Less than high school'}: & \quad 195 \times (22.47 - 23.33)^2 \\
& = 195 \times (-0.86)^2 \\
& = 195 \times 0.7396 \\
& = 144.222 \\
\text{For 'High school'}: & \quad 590 \times (22.87 - 23.33)^2 \\
& = 590 \times (-0.46)^2 \\
& = 590 \times 0.2116 \\
& = 124.844 \\
\text{For 'Junior college'}: & \quad 86 \times (23.71 - 23.33)^2 \\
& = 86 \times 0.38^2 \\
& = 86 \times 0.1444 \\
& = 12.4184 \\
\text{For 'Bachelor\'s degree'}: & \quad 185 \times (24.58 - 23.33)^2 \\
& = 185 \times 1.25^2 \\
& = 185 \times 1.5625 \\
& = 289.0625 \\
\text{For 'Graduate degree'}: & \quad 104 \times (25.10 - 23.33)^2 \\
& = 104 \times 1.77^2 \\
& = 104 \times 3.1329 \\
& = 325.8226 \\
\end{align*}
\][/tex]
4. Sum the contributions from all groups:
[tex]\[
SS_{\text{between}} = 144.222 + 124.844 + 12.4184 + 289.0625 + 325.8226 = 896.3685
\][/tex]
5. Result:
The calculated value for the between-group sum of squares (SS_between) is approximately 896.3685.
This value reflects the total variation among the group means relative to the overall mean. This is an essential step for further analysis, such as conducting an ANOVA test to determine if the observed differences in group means are statistically significant.
