Below are the jersey numbers of 11 players randomly selected from a football team. Find the​ range, variance, and standard deviation for the given sample data. What do the results tell​ us?
24   6   17   20   76   3   26   13   40   21   57

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!

Learn how to calculate the range, variance, and standard deviation for a sample set of football jersey numbers. This problem provides a step-by-step solution, including the calculation of the mean, variance, and standard deviation for the given data. Suitable for students in Grades 10-12.

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