Find the sample variance and standard deviation.
4​, 55​, 15​, 48​, 32​, 24​, 33​, 31​, 31​, 32   
Question content area bottom
Part 1
Choose the correct answer below. Fill in the answer box to complete your choice.
​(Round to two decimal places as​ needed.)
A.sigma squared equals enter your response here
sigma squaredequals
  
enter your response here
B.s squared equals 211.40
s squaredequals
  
211.40
Your answer is correct.
Part 2
Choose the correct answer below. Fill in the answer box to complete your choice.
​(Round to one decimal place as​ needed.)
A.sigma equals enter your response here
sigmaequals
  
enter your response here
B.
sequals
  
enter your response here 

To find the sample variance and standard deviation of a set of numbers, we follow these steps:

### Step 1: Calculate the Mean
The mean \((\bar{x})\) of the data set is calculated by summing all the numbers and dividing by the total number of values.

\[
\bar{x} = \frac{\sum x_i}{n}
\]

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

Given the dataTo find the sample variance and standard deviation of a set of numbers, we follow these steps:

### Step 1: Calculate the Mean
The mean \((\bar{x})\) of the data set is calculated by summing all the numbers and dividing by the total number of values.

\[
\bar{x} = \frac{\sum x_i}{n}
\]

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

Given the  \(4, 55, 15, 48, 32, 24, 33, 31, 31, 32\),

\[
\bar{x} = \frac{4 + 55 + 15 + 48 + 32 + 24 + 33 + 31 + 31 + 32}{10}
\]

### Step 2: Calculate the Sample Variance
The sample variance \(s^2\) is calculated using the formula:

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

### Step 3: Calculate the Sample Standard Deviation
The sample standard deviation \(s\) is the square root of the variance:

\[
s = \sqrt{s^2}
\]

Let's calculate these values step-by-step.### Results:

1. **Mean (\(\bar{x}\))**: \(30.5\)
2. **Sample Variance (\(s^2\))**: \(211.39\) (rounded to two decimal places)
3. **Sample Standard Deviation (\(s\))**: \(14.5\) (rounded to one decimal place)

### Answers:
- **Part 1:**
  - The sample variance \(s^2 = 211.40\).
- **Part 2:**
  - The sample standard deviation \(s = 14.5\).

If you have any questions or would like a further explanation, please let me know!

### Related Questions:
1. What is the difference between sample variance and population variance?
2. How does the sample size affect the variance and standard deviation?
3. Can variance ever be negative? Why or why not?
4. What are some real-world examples where variance and standard deviation are used?
5. How can outliers impact the mean, variance, and standard deviation?

**Tip:** Remember that the standard deviation is a measure of the spread of data points from the mean; a higher standard deviation indicates more spread.

Find the sample variance and standard deviation.
4, 55, 15, 48, 32, 24, 33, 31, 31, 32
Question content area bottom
Part 1
Choose the correct answer below. Fill in the answer box to complete your choice.
(Round to two decimal places as needed.)
A. sigma squared equals enter your response here
B. s squared equals 211.40
Your answer is correct.
Part 2
Choose the correct answer below. Fill in the answer box to complete your choice.
(Round to one decimal place as needed.)
A. sigma equals enter your response here
B. s equals enter your response here

Learn how to calculate the sample variance and standard deviation for the data set 4, 55, 15, 48, 32, 24, 33, 31, 31, 32 with a detailed step-by-step solution. This problem covers key statistical concepts including variance and standard deviation, suitable for students in Grades 9-12.

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