https://file.aiquickdraw.com/mathbot/file/17297796679653rrwnziv.jpg

The problem asks you to determine the range, mean, variance, and standard deviation of a dataset representing the number of wins for 10 football teams in a season. The dataset is as follows:

\[
2, 9, 15, 2, 15, 10, 13, 7, 4, 10
\]

### Step 1: Range
The range is the difference between the maximum and minimum values in the dataset:
- Maximum value: 15
- Minimum value: 2

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

This confirms that the range is **13**, which is already provided.

### Step 2: Population Mean
The mean (or average) is calculated by summing all the values and dividing by the number of values (10 in this case).

\[
\text{Mean} = \frac{2 + 9 + 15 + 2 + 15 + 10 + 13 + 7 + 4 + 10}{10} = \frac{87}{10} = 8.7
\]

The population mean is **8.7** (rounded to the nearest tenth).

### Step 3: Variance
The population variance is the average of the squared differences from the mean. First, calculate the squared differences for each value:

\[
\begin{align*}
(2 - 8.7)^2 & = (-6.7)^2 = 44.89 \\
(9 - 8.7)^2 & = (0.3)^2 = 0.09 \\
(15 - 8.7)^2 & = (6.3)^2 = 39.69 \\
(2 - 8.7)^2 & = (-6.7)^2 = 44.89 \\
(15 - 8.7)^2 & = (6.3)^2 = 39.69 \\
(10 - 8.7)^2 & = (1.3)^2 = 1.69 \\
(13 - 8.7)^2 & = (4.3)^2 = 18.49 \\
(7 - 8.7)^2 & = (-1.7)^2 = 2.89 \\
(4 - 8.7)^2 & = (-4.7)^2 = 22.09 \\
(10 - 8.7)^2 & = (1.3)^2 = 1.69 \\
\end{align*}
\]

Now, sum these squared differences:

\[
44.89 + 0.09 + 39.69 + 44.89 + 39.69 + 1.69 + 18.49 + 2.89 + 22.09 + 1.69 = 216.1
\]

Divide by 10 (since it's a population variance):

\[
\text{Variance} = \frac{216.1}{10} = 21.61
\]

### Step 4: Standard Deviation
The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{21.61} \approx 4.65
\]

### Summary:
- **Range**: 13
- **Mean**: 8.7
- **Variance**: 21.61
- **Standard Deviation**: 4.65

Would you like any more details or explanations?

Here are 5 follow-up questions you might find useful:
1. How would the calculations change if this were a sample rather than a population?
2. Can you describe the relationship between variance and standard deviation?
3. What is the interpretation of the range in the context of this dataset?
4. How would you calculate the median for this data set?
5. How does the mean compare to the mode of the dataset?

**Tip**: Always double-check whether you're working with a sample or population since it affects how variance is calculated (dividing by \( n-1 \) for a sample).

The numbers of regular season wins for 10 football teams in a given season are: 2, 9, 15, 2, 15, 10, 13, 7, 4, 10. Determine the range, mean, variance, and standard deviation of the population data set.

This problem involves calculating the range, mean, variance, and standard deviation of football team wins data. The dataset contains the number of wins for 10 teams in a season, and includes key statistical concepts suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation