Calculating the Variance among Sample Means late the variance among our four means otherwise known as between-columnt we use the formular: \( \hat{\sigma}_{\text {Lawern }}^{2}=\frac{n_{1}\left(\bar{X}_{1}-\bar{X}\right)^{2}+n_{2}\left(\bar{X}_{2}-\bar{X}\right)^{2}+\ldots n k\left(\bar{X}_{k}-\bar{X}\right)^{2}}{k-1}=\sum_{i=1}^{k} \frac{n_{1}\left(X_{i}-\bar{X}\right)^{2}}{K-1} \) \( n_{i}(i=1 \ldots k)= \) number of items or observations in samples \( 1,2, \ldots \mathrm{k} \) iumber of sample being studied.

To calculate the variance among sample means (also known as the between-group variance), you can use the formula you provided, which is a form of the formula for the between-group variance in ANOVA (Analysis of Variance). Let's clarify and correct the notation and the formula for better understanding. ### Formula for Between-Group Variance The formula for the variance among sample means (between-group variance) is given by: \[ \hat{\sigma}_{\text{between}}^{2} = \frac{\sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X})^2}{k - 1} \] Where: - \( k \) = number of groups (samples) - \( n_i \) = number of observations in group \( i \) - \( \bar{X}_i \) = mean of group \( i \) - \( \bar{X} \) = overall mean of all groups combined ### Steps to Calculate the Between-Group Variance 1. **Calculate the Overall Mean (\( \bar{X} \))**: \[ \bar{X} = \frac{\sum_{i=1}^{k} n_i \bar{X}_i}{\sum_{i=1}^{k} n_i} \] 2. **Calculate the Between-Group Variance**: - For each group, calculate the squared difference between the group mean (\( \bar{X}_i \)) and the overall mean (\( \bar{X} \)). - Multiply each squared difference by the number of observations in that group (\( n_i \)). - Sum these values across all groups. - Divide by \( k - 1 \) (where \( k \) is the number of groups). ### Example Calculation Suppose you have the following sample means and sizes: - Group 1: \( n_1 = 5 \), \( \bar{X}_1 = 10 \) - Group 2: \( n_2 = 5 \), \( \bar{X}_2 = 15 \) - Group 3: \( n_3 = 5 \), \( \bar{X}_3 = 20 \) 1. **Calculate the Overall Mean**: \[ \bar{X} = \frac{(5 \times 10) + (5 \times 15) + (5 \times 20)}{5 + 5 + 5} = \frac{50 + 75 + 100}{15} = \frac{225}{15} = 15 \] 2. **Calculate the Between-Group Variance**: \[ \hat{\sigma}_{\text{between}}^{2} = \frac{5(10 - 15)^2 + 5(15 - 15)^2 + 5(20 - 15)^2}{3 - 1} \] \[ = \frac{5(25) + 5(0) + 5(25)}{2} = \frac{125 + 0 + 125}{2} = \frac{250}{2} = 125 \] Thus, the variance among the sample means (between-group variance) is \( 125 \). ### Conclusion This method allows you to quantify the variability among the means of different groups, which is essential in statistical analyses such as ANOVA. If you have specific data or groups you want to analyze, feel free to provide that, and I can help you with the calculations!