Let's break down the calculations step by step:

Step 1: Compute the Range

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

• Maximum value = 76
• Minimum value = 3

Range = 76 - 3 = 73

Step 2: Compute the Variance and Standard Deviation

Since we're working with a sample (not the entire population), we'll use the formula for sample variance and sample standard deviation.

• Sample variance formula: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ where $x_i$ are the data points, $\bar{x}$ is the mean of the sample, and $n$ is the number of data points.

• Sample standard deviation is simply the square root of the variance: $s = \sqrt{s^2}$

Step 3: Calculate the Mean

To find the mean ($\bar{x}$) of the data, sum all the values and divide by the number of data points.

Given dataLet's break down the calculations step by step:

Step 1: Compute the Range

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

• Maximum value = 76
• Minimum value = 3

Range = 76 - 3 = 73

Step 2: Compute the Variance and Standard Deviation

Since we're working with a sample (not the entire population), we'll use the formula for sample variance and sample standard deviation.

• Sample variance formula: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ where $x_i$ are the data points, $\bar{x}$ is the mean of the sample, and $n$ is the number of data points.

• Sample standard deviation is simply the square root of the variance: $s = \sqrt{s^2}$

Step 3: Calculate the Mean

To find the mean ($\bar{x}$) of the data, sum all the values and divide by the number of data points.

Given 24, 6, 17, 20, 76, 3, 26, 13, 40, 21, 57

$\bar{x} = \frac{24 + 6 + 17 + 20 + 76 + 3 + 26 + 13 + 40 + 21 + 57}{11} = \frac{303}{11} = 27.545$

Step 4: Calculate the Variance

Now, we calculate the variance by finding the squared differences from the mean:

$s^2 = \frac{(24 - 27.545)^2 + (6 - 27.545)^2 + \ldots + (57 - 27.545)^2}{11-1}$

I'll compute this next.Here are the computed results:

• Mean: 27.55
• Range: 73
• Sample Variance: 487.47
• Sample Standard Deviation: 22.08

Interpretation:

• Range: The range of 73 indicates that there is a wide spread between the smallest (3) and largest (76) jersey numbers.
• Variance and Standard Deviation: The standard deviation of 22.08 and variance of 487.47 suggest that the jersey numbers are spread out around the mean (27.55), showing moderate variability within the data.

Would you like more details on any part of this process?

Related Questions:

1. How do the variance and standard deviation change if we add or remove data points?
2. What is the importance of distinguishing between population and sample variance?
3. How would the results change if all jersey numbers were closer together?
4. How can standard deviation help in comparing two different sets of jersey numbers?
5. Can outliers, like 76, significantly affect the variance and standard deviation?

Tip:

Always check for outliers, as they can dramatically affect the range, variance, and standard deviation!