Let's analyze the question from the image:

The problem provides the following mileage values for five tires:
17,000; 18,000; 18,000; 21,000; 21,000.

We are tasked with finding:

1. The mean.
2. The range.
3. The standard deviation, rounded to the nearest unit.

Step 1: Calculate the Mean

The mean is the average of the data values:
$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$
$\text{Mean} = \frac{17,000 + 18,000 + 18,000 + 21,000 + 21,000}{5}$
$\text{Mean} = \frac{95,000}{5} = 19,000$

So, Mean = 19,000 miles.

Step 2: Calculate the Range

The range is the difference between the largest and smallest values:
$\text{Range} = \text{Largest value} - \text{Smallest value}$
$\text{Range} = 21,000 - 17,000 = 4,000$

So, Range = 4,000 miles.

Step 3: Calculate the Standard Deviation

The standard deviation measures the spread of the data. The formula is:
$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$
Where:

• $x_i$ are the data points,
• $\mu$ is the mean,
• $N$ is the number of data points.

Step 3.1: Calculate deviations from the mean and square them

The mean ($\mu$) is 19,000. For each $x_i$:
$(17,000 - 19,000)^2 = (-2,000)^2 = 4,000,000$
$(18,000 - 19,000)^2 = (-1,000)^2 = 1,000,000$
$(18,000 - 19,000)^2 = (-1,000)^2 = 1,000,000$
$(21,000 - 19,000)^2 = (2,000)^2 = 4,000,000$
$(21,000 - 19,000)^2 = (2,000)^2 = 4,000,000$

Step 3.2: Sum of squared deviations

$\text{Sum} = 4,000,000 + 1,000,000 + 1,000,000 + 4,000,000 + 4,000,000 = 14,000,000$

Step 3.3: Divide by $N = 5$

$\text{Variance} = \frac{\text{Sum}}{N} = \frac{14,000,000}{5} = 2,800,000$

Step 3.4: Take the square root

$\sigma = \sqrt{2,800,000} \approx 1,673$

So, Standard Deviation = 1,673 miles (rounded to the nearest unit).

Final Answers:

1. Mean = 19,000 miles ✅
2. Range = 4,000 miles ✅
3. Standard Deviation = 1,673 miles ❌ (in the image, this might not have been entered correctly).

Do you want me to clarify any part of this solution? 😊

Follow-Up Questions:

1. What does the standard deviation tell us about the spread of the mileage values?
2. How would the mean change if one value (e.g., 17,000) were replaced by 16,000?
3. What is the effect of adding a constant to all data points on the range and standard deviation?
4. Can we calculate the population standard deviation differently for larger data sets?
5. Why is it important to round the standard deviation to the nearest unit in this context?

Tip:

When working with standard deviation, ensure all calculations are accurate to avoid rounding errors early in the process!