The numbers of regular season wins for 10 football teams in a given season are given below. Determine the​ range, mean,​ variance, and standard deviation of the population data set. 
2​, 7​, 15​, 4​, 11​, 8​, 11​, 6​, 4​, 9   
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Part 1
The range is 
  
13. 
​(Simplify your​ answer.)
Part 2
The population mean is 
  
enter your response here. 
​(Simplify your answer. Round to the nearest tenth as​ needed.)

Let's start by calculating each of the required statistical measures: the range, mean, variance, and standard deviation.

### **Given Data:**
The numbers of regular season wins for the 10 football teams are:
\[ 2, 7, 15, 4, 11, 8, 11, 6, 4, 9 \]

### **Part 1: Range**
The range of a data set is calculated as the difference between the maximum and minimum values.

- **Minimum value**: \(2\)
- **Maximum value**: \(15\)

\[
\text{Range} = \text{Maximum} - \text{Minimum} = 15 - 2 = 13
\]

So, the range is **13**.

### **Part 2: Population Mean**
The population mean is calculated by summing all the values and dividing by the number of values.

\[
\text{Mean} = \frac{\sum X_i}{N} = \frac{2 + 7 + 15 + 4 + 11 + 8 + 11 + 6 + 4 + 9}{10}
\]

\[
\text{Mean} = \frac{77}{10} = 7.7
\]

So, the population mean is **7.7**.

### **Part 3: Population Variance**
The population variance is calculated using the formula:

\[
\text{Variance} = \frac{\sum (X_i - \mu)^2}{N}
\]

Where \(\mu\) is the population mean.

First, let's find \((X_i - \mu)^2\) for each value:

- \( (2 - 7.7)^2 = 32.49 \)
- \( (7 - 7.7)^2 = 0.49 \)
- \( (15 - 7.7)^2 = 53.29 \)
- \( (4 - 7.7)^2 = 13.69 \)
- \( (11 - 7.7)^2 = 10.89 \)
- \( (8 - 7.7)^2 = 0.09 \)
- \( (11 - 7.7)^2 = 10.89 \)
- \( (6 - 7.7)^2 = 2.89 \)
- \( (4 - 7.7)^2 = 13.69 \)
- \( (9 - 7.7)^2 = 1.69 \)

Summing these squared deviations:

\[
\sum (X_i - \mu)^2 = 140.1
\]

Now, divide by the population size \(N = 10\):

\[
\text{Variance} = \frac{140.1}{10} = 14.01
\]

So, the population variance is **14.01**.

### **Part 4: Population Standard Deviation**
The population standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{14.01} \approx 3.7
\]

So, the population standard deviation is **3.7**.

### **Summary**
- **Range**: \(13\)
- **Mean**: \(7.7\)
- **Variance**: \(14.01\)
- **Standard Deviation**: \(3.7\)

Would you like further details or have any other questions?

Here are some related questions:
1. How would the standard deviation change if a new value of 20 was added?
2. Can you calculate the median of this data set?
3. What is the mode of this data set?
4. How does the variance compare if this data set was treated as a sample instead of a population?
5. How would the mean change if the value 4 was removed?

**Tip:** Always check whether the dataset is a sample or a population when calculating variance and standard deviation, as the formulas differ slightly.

The numbers of regular season wins for 10 football teams in a given season are given below. Determine the range, mean, variance, and standard deviation of the population data set. 2, 7, 15, 4, 11, 8, 11, 6, 4, 9

Learn how to calculate the range, mean, variance, and standard deviation for a population data set of football team wins. This problem involves descriptive statistics concepts and key formulas. Suitable for students in Grades 9-12.

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