The image provided is from a statistics textbook, focusing on the computation of the Total Sum of Squares (SST) for Armand's Pizza Parlors. Here is a breakdown and explanation of the contents:

1. Table 14.4 Overview:

   • The table shows data on the student population $x_i$ (in thousands) and quarterly sales $y_i$ (in thousands of dollars) for 10 pizza parlors (restaurants).
   • It calculates the deviation of each $y_i$ value from the mean quarterly sales $\bar{y} = 130$, and the squared deviation, $(y_i - \bar{y})^2$.

2. SST (Total Sum of Squares):

   • The goal is to measure the total variation in $y$ values from the mean.
   • Formula for SST: $\text{SST} = \sum (y_i - \bar{y})^2$.
   • In this case, $\bar{y}$ (mean sales) is calculated as $130$, based on a total sales sum of $1300$ across 10 restaurants.
   • Summing all the squared deviations yields $\text{SST} = 15,730$, which represents the overall variability in sales from the mean.

3. Clarification of Terms:

   • SST (Total Sum of Squares): This is the measure of total variability in $y$ (sales) values around the mean $\bar{y}$.
   • SSE (Sum of Squared Errors): Although not directly shown in this section, SSE would measure the deviation of observed values $y_i$ from predicted values $\hat{y}_i$ (from a regression model).
   • SSR (Sum of Squares due to Regression): This measures the variability explained by the regression model, specifically the deviation of predicted values $\hat{y}_i$ from the mean $\bar{y}$.

4. Example Calculations in the Table:

   • For each restaurant, the deviation $y_i - \bar{y}$ is calculated. For example, for restaurant 1, $y_1 = 58$ and $\bar{y} = 130$, so $y_1 - \bar{y} = -72$.
   • The squared deviation, $(y_i - \bar{y})^2$, for restaurant 1 is $(-72)^2 = 5,184$.
   • This is done for each restaurant, and the values are summed to get the SST.

5. Interpretation of SST:

   • SST represents the overall dispersion of sales values around the mean sales. A higher SST indicates greater variability in sales across restaurants.

Would you like further clarification on any calculations or concepts?

5 Related Questions

1. How is the mean ($\bar{y}$) calculated in this example?
2. What is the difference between SST and SSE?
3. How would SST change if the mean value of sales ($\bar{y}$) was different?
4. What is the purpose of calculating SSR and how does it relate to SST and SSE?
5. Why are squared deviations used instead of just deviations?

Tip

When calculating sums of squares in regression, remember: SST = SSR + SSE. This breakdown helps assess how much of the total variation in data is explained by the model versus unexplained (errors).