To solve for the range and sample standard deviation, we will follow these steps:

Step 1: Calculate the Range

The range is the difference between the maximum and minimum values in the dataset.

• Minimum value: $2$
• Maximum value: $198$

$\text{Range} = 198 - 2 = 196$

Step 2: Calculate the Sample Standard Deviation

The formula for sample standard deviation ($s$) is:

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$

Where:

• $x_i$ represents each data point,
• $\bar{x}$ is the sample mean,
• $n$ is the sample size.

Step 2.1: Find the Mean

First, we calculate the mean ($\bar{x}$) of the dataTo solve for the range and sample standard deviation, we will follow these steps:

Step 1: Calculate the Range

The range is the difference between the maximum and minimum values in the dataset.

• Minimum value: $2$
• Maximum value: $198$

$\text{Range} = 198 - 2 = 196$

Step 2: Calculate the Sample Standard Deviation

The formula for sample standard deviation ($s$) is:

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$

Where:

• $x_i$ represents each data point,
• $\bar{x}$ is the sample mean,
• $n$ is the sample size.

Step 2.1: Find the Mean

First, we calculate the mean ($\bar{x}$) of the

$\bar{x} = \frac{4 + 2 + 52 + 114 + 198 + 95 + 66 + 86 + 72 + 60 + 103 + 92}{12}$

Let's compute the mean:

$\bar{x} = \frac{944}{12} = 78.67$

Step 2.2: Find the Squared Differences

Now, we calculate the squared differences between each data point and the mean:

$(4 - 78.67)^2 = 5585.35$ $(2 - 78.67)^2 = 5886.35$ $(52 - 78.67)^2 = 711.89$ $(114 - 78.67)^2 = 1210.89$ $(198 - 78.67)^2 = 14316.69$ $(95 - 78.67)^2 = 263.39$ $(66 - 78.67)^2 = 163.39$ $(86 - 78.67)^2 = 53.79$ $(72 - 78.67)^2 = 44.49$ $(60 - 78.67)^2 = 351.29$ $(103 - 78.67)^2 = 589.69$ $(92 - 78.67)^2 = 177.69$

Step 2.3: Compute the Sum of Squared Differences

Next, we sum these squared differences:

$\text{Sum} = 5585.35 + 5886.35 + 711.89 + 1210.89 + 14316.69 + 263.39 + 163.39 + 53.79 + 44.49 + 351.29 + 589.69 + 177.69$ $\text{Sum} = 29054.9$

Step 2.4: Divide by $n - 1$

Since this is a sample, we divide by $n - 1 = 12 - 1 = 11$:

$\frac{29054.9}{11} = 2641.35$

Step 2.5: Take the Square Root

Finally, we take the square root to find the sample standard deviation:

$s = \sqrt{2641.35} \approx 51.39$

Final Answers:

• Range: 196
• Sample Standard Deviation: 51.39

Would you like more details or have any questions about these calculations?

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Here are 5 relative questions that expand this information:

1. How does the range affect the spread of the data?
2. What is the difference between sample standard deviation and population standard deviation?
3. How would the standard deviation change if the maximum value increased?
4. Can you interpret what a high standard deviation means in this context?
5. How does the sample size influence the accuracy of the standard deviation?

Tip: Always verify whether you're calculating the sample or population standard deviation, as this changes the divisor in the formula.