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.