https://file.aiquickdraw.com/mathbot/file/1729446908761bi1p2lau.jpg

Let's solve the problem step-by-step based on the image provided. The given sample data set is:

\[
x = \{26.8, 5.7, 8.6, 21, 24.1, 23.6, 27.1\}
\]

### Part (a) - Finding the Mean:
The formula for the mean (average) of a data set is:

\[
\text{Mean} = \frac{\sum{x_i}}{n}
\]

where \(n\) is the number of data points.

1. Sum of the dataLet's solve the problem step-by-step based on the image provided. The given sample data set is:

\[
x = \{26.8, 5.7, 8.6, 21, 24.1, 23.6, 27.1\}
\]

### Part (a) - Finding the Mean:
The formula for the mean (average) of a data set is:

\[
\text{Mean} = \frac{\sum{x_i}}{n}
\]

where \(n\) is the number of data points.

1. Sum of the 
\[
26.8 + 5.7 + 8.6 + 21 + 24.1 + 23.6 + 27.1 = 136.9
\]

2. Number of data points: \(n = 7\)

3. Calculate the mean:
\[
\text{Mean} = \frac{136.9}{7} = 19.56
\]

So, the **mean** of the data is:
\[
\boxed{19.56}
\]

---

### Part (b) - Finding the Sample Standard Deviation:
The formula for the **sample standard deviation** is:

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

where:
- \(x_i\) are the data points,
- \(\bar{x}\) is the mean (19.56),
- \(n\) is the number of data points (7).

#### Step 1: Calculate the squared differences from the mean.

\[
(x_i - \bar{x})^2
\]

- \( (26.8 - 19.56)^2 = 52.49 \)
- \( (5.7 - 19.56)^2 = 192.34 \)
- \( (8.6 - 19.56)^2 = 120.34 \)
- \( (21 - 19.56)^2 = 2.07 \)
- \( (24.1 - 19.56)^2 = 20.60 \)
- \( (23.6 - 19.56)^2 = 16.39 \)
- \( (27.1 - 19.56)^2 = 56.71 \)

#### Step 2: Sum the squared differences.
\[
52.49 + 192.34 + 120.34 + 2.07 + 20.60 + 16.39 + 56.71 = 460.94
\]

#### Step 3: Divide by \(n - 1 = 6\).
\[
\frac{460.94}{6} = 76.82
\]

#### Step 4: Take the square root of the result.
\[
s = \sqrt{76.82} = 8.76
\]

So, the **sample standard deviation** is:
\[
\boxed{8.76}
\]

---

### Final Answers:
- **Mean**: \(19.56\)
- **Sample Standard Deviation**: \(8.76\)

Do you have any questions or need further clarification?

---

### 5 Related Questions:
1. What is the difference between sample and population standard deviation?
2. How would the mean change if a new data point (e.g., 10) was added?
3. How can you calculate variance from the standard deviation?
4. What are some real-life examples of using standard deviation?
5. How does removing an outlier affect the standard deviation?

### Tip:
Always use the sample standard deviation formula when working with smaller datasets, as it adjusts for the fact that you are not measuring the entire population.

Find the mean and the sample standard deviation for the following data set: 26.8, 5.7, 8.6, 21, 24.1, 23.6, 27.1.

Learn how to find the mean and sample standard deviation for the data set: 26.8, 5.7, 8.6, 21, 24.1, 23.6, 27.1. This detailed solution involves key statistical concepts and step-by-step calculations, perfect for high school students.

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