To calculate the mean square for error (MSE), we use the following formula:

$MSE = \frac{\text{Sum of Squares for Error (SSE)}}{\text{Degrees of Freedom for Error (df)}}$

From the image, we have the sum of squares for error (SSE) = 17.7685.

To find the degrees of freedom for error ($df$), we use:

$df = N - k$

Where:

• $N$ is the total number of observations.
• $k$ is the number of groups (or asset classes).

Step 1: Count the total number of observations $N$

From the table, we have:

• Small Cap: 8 funds.
• Mid Cap: 7 funds.
• Large Cap: 8 funds.

So, the total number of observations $N = 8 + 7 + 8 = 23$.

Step 2: Determine the number of groups $k$

There are 3 asset classes (Small Cap, Mid Cap, and Large Cap), so $k = 3$.

Step 3: Calculate the degrees of freedom for error ($df$)

$df = N - k = 23 - 3 = 20$

Step 4: Calculate the mean square for error (MSE)

$MSE = \frac{17.7685}{20} = 0.888425$

Rounding to two decimal places:

$MSE \approx 0.89$

Thus, the mean square for error is approximately 0.89.

Do you have any further questions or would you like more details on this?

Here are five related questions:

1. How is the sum of squares for error (SSE) calculated?
2. What is the difference between the sum of squares for treatments and error?
3. Why do we subtract the number of groups from the total observations to calculate degrees of freedom?
4. How would you interpret the MSE in the context of this problem?
5. What is the next step after calculating MSE in an ANOVA test?

Tip: MSE is a key metric in ANOVA (Analysis of Variance), helping assess how much variation in data is due to random error versus treatment effects.