Let's first find the sample mean and the sample standard deviation for the given data set:

Data Set:
45, 55, 53, 56, 31, 31, 45, 55, 53, 33, 56, 45, 48

1. Sample Mean Calculation:

The formula for the sample mean is:

$\bar{x} = \frac{\sum{x_i}}{n}$

Where:

• $\sum{x_i}$ is the sum of all data points.
• $n$ is the number of data points.

Sum of the data points: $45 + 55 + 53 + 56 + 31 + 31 + 45 + 55 + 53 + 33 + 56 + 45 + 48 = 601$

Number of data points ($n$): $n = 13$

So, the sample mean is:

$\bar{x} = \frac{601}{13} = 46.23 \, (\text{rounded to 2 decimal places})$

2. Sample Standard Deviation Calculation:

The formula for the sample standard deviation is:

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

Where:

• $x_i$ are the individual data points.
• $\bar{x}$ is the sample mean.
• $n$ is the number of data points.

Now, let's compute each of the squared differences:

1. $(45 - 46.23)^2 = (-1.23)^2 = 1.5129$
2. $(55 - 46.23)^2 = (8.77)^2 = 76.6929$
3. $(53 - 46.23)^2 = (6.77)^2 = 45.7929$
4. $(56 - 46.23)^2 = (9.77)^2 = 95.1929$
5. $(31 - 46.23)^2 = (-15.23)^2 = 232.0329$
6. $(31 - 46.23)^2 = (-15.23)^2 = 232.0329$
7. $(45 - 46.23)^2 = (-1.23)^2 = 1.5129$
8. $(55 - 46.23)^2 = (8.77)^2 = 76.6929$
9. $(53 - 46.23)^2 = (6.77)^2 = 45.7929$
10. $(33 - 46.23)^2 = (-13.23)^2 = 174.0529$
11. $(56 - 46.23)^2 = (9.77)^2 = 95.1929$
12. $(45 - 46.23)^2 = (-1.23)^2 = 1.5129$
13. $(48 - 46.23)^2 = (1.77)^2 = 3.1329$

Now, sum these squared differences:

$1.5129 + 76.6929 + 45.7929 + 95.1929 + 232.0329 + 232.0329 + 1.5129 + 76.6929 + 45.7929 + 174.0529 + 95.1929 + 1.5129 + 3.1329 = 1079.1863$

Now, divide by $n - 1 = 13 - 1 = 12$:

$\frac{1079.1863}{12} = 89.9322$

Finally, take the square root:

$s = \sqrt{89.9322} = 9.49 \, (\text{rounded to 2 decimal places})$

Final Results:

• Sample mean: 46.23
• Sample standard deviation: 9.49

Would you like further details or clarifications? Here are some related questions:

1. What would happen to the sample standard deviation if we added more data points?
2. How is the sample standard deviation different from the population standard deviation?
3. How do you calculate the variance from the standard deviation?
4. What is the significance of the sample mean in statistics?
5. Can you apply the formula for standard deviation to a population instead of a sample?

Tip: When working with sample statistics, remember that we divide by $n-1$ instead of $n$ to correct for the bias in estimating the population parameters.