tending Concepis Altermative Formula You used SS, $$ =\Sigma(x-x)^{2} $$ when ealculating varianes and standard deviation. An alternative formula that is sometimes mere conveniont for hand calculations is $$ S S_{x}=\Sigma \Sigma x^{2}-\frac{(\Sigma x)^{2}}{11} $$ You can find the sample variance by dividing the sum of squares by $$ 1-1 $$ and the sample standard deviation by finding the square root of the ample variance. a) Show how to obtain the alternative formula, b) Use the alternative formula to calculate the sample standard deviation for the data set in Exercise 15, 2) Compare your result with the sample standard deviation obtained in Exercise 15.

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Certainly! Let's address each part of your query step by step.

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### **a) Deriving the Alternative Formula for Sum of Squares ($$ SS_x $$)**

The **Sum of Squares** ($$ SS_x $$) is a fundamental component in calculating both variance and standard deviation. The traditional formula is:

$$
SS_x = \Sigma (x_i - \bar{x})^2
$$

where:
- $$ x_i $$ represents each data point.
- $$ \bar{x} $$ is the mean of the data set.

However, there's an alternative formula that's often more convenient for manual calculations:

$$
SS_x = \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}
$$

**Let's derive this alternative formula:**

1. **Start with the Traditional Formula:**

$$
SS_x = \Sigma (x_i - \bar{x})^2
$$

2. **Expand the Squared Term:**

$$
SS_x = \Sigma (x_i^2 - 2x_i \bar{x} + \bar{x}^2)
$$

3. **Distribute the Summation:**

$$
SS_x = \Sigma x_i^2 - 2\bar{x} \Sigma x_i + \Sigma \bar{x}^2
$$

4. **Simplify Each Term:**
   - $$ \Sigma x_i $$ is the sum of all data points, which is $$ n\bar{x} $$ (since $$ \bar{x} = \frac{\Sigma x_i}{n} $$).
   - $$ \Sigma \bar{x}^2 = n \bar{x}^2 $$ because you're summing the constant $$ \bar{x}^2 $$ $$ n $$ times.

5. **Substitute Simplified Terms Back:**

$$
SS_x = \Sigma x_i^2 - 2\bar{x} (n\bar{x}) + n\bar{x}^2
$$

$$
SS_x = \Sigma x_i^2 - 2n\bar{x}^2 + n\bar{x}^2
$$

$$
SS_x = \Sigma x_i^2 - n\bar{x}^2
$$

6. **Express $$ \bar{x} $$ in Terms of $$ \Sigma x_i $$:**

$$
\bar{x} = \frac{\Sigma x_i}{n} \quad \Rightarrow \quad \bar{x}^2 = \left(\frac{\Sigma x_i}{n}\right)^2 = \frac{(\Sigma x_i)^2}{n^2}
$$

7. **Substitute Back into $$ SS_x $$:**

$$
SS_x = \Sigma x_i^2 - n \left( \frac{(\Sigma x_i)^2}{n^2} \right )
$$

$$
SS_x = \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}
$$

**Thus, we've derived the alternative formula:**

$$
SS_x = \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}
$$

---

### **b) Calculating the Sample Standard Deviation Using the Alternative Formula**

To compute the **sample standard deviation** using the alternative formula, follow these steps:

1. **Compute $$ \Sigma x_i $$:** Sum all data points.
2. **Compute $$ \Sigma x_i^2 $$:** Sum the squares of all data points.
3. **Apply the Alternative $$ SS_x $$ Formula:**

$$
SS_x = \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}
$$

4. **Calculate the Sample Variance ($$ s^2 $$):**

$$
s^2 = \frac{SS_x}{n - 1}
$$

5. **Compute the Sample Standard Deviation ($$ s $$):**

$$
s = \sqrt{s^2} = \sqrt{\frac{\Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}}{n - 1}}
$$

**However, to perform the actual calculation, I need the specific data set from **Exercise 15**.** Please provide the data points from Exercise 15, and I'll gladly carry out the computation for you.

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### **2) Comparing with the Sample Standard Deviation Obtained in Exercise 15**

Once you've provided the data set from Exercise 15, after performing the calculation in part (b), we can compare the result with the sample standard deviation obtained previously in that exercise to ensure consistency and verify the correctness of both methods.

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**Feel free to share the data set from Exercise 15, and I'll assist you further with the calculations!**
To find the sample standard deviation using the alternative formula: 1. **Sum all data points ($$ \Sigma x_i $$)** 2. **Sum the squares of all data points ($$ \Sigma x_i^2 $$)** 3. **Apply the formula:** $$ SS_x = \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n} $$ 4. **Calculate the sample variance:** $$ s^2 = \frac{SS_x}{n - 1} $$ 5. **Find the sample standard deviation:** $$ s = \sqrt{s^2} $$ After performing these steps with the data from Exercise 15, compare the result with the standard deviation obtained earlier to ensure consistency.

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