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.