Let's calculate each of the requested statistical measures for the given highway fuel economy values: $20, 23, 26, 27, 30, 31, 35, 35$.

1. Mean

The mean is the average of the values.

$\text{Mean} = \frac{\sum \text{values}}{\text{number of values}} = \frac{20 + 23 + 26 + 27 + 30 + 31 + 35 + 35}{8}$

$\text{Mean} = \frac{227}{8} = 28.4$

2. Median

The median is the middle value when the data is ordered. If there is an even number of observations, the median is the average of the two middle numbers.

The ordered data is: $20, 23, 26, 27, 30, 31, 35, 35$.

Since there are 8 values (an even number), the median is the average of the 4th and 5th values:

$\text{Median} = \frac{27 + 30}{2} = \frac{57}{2} = 28.5$

3. Mode

The mode is the value that appears most frequently in the data set.

Here, the number $35$ appears twice, which is more than any other number.

$\text{Mode} = 35$

4. Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of values.

Step 1: Find the mean (already calculated as 28.4).

Step 2: Subtract the mean from each value, and then square the result.

(20 - 28.4)^2 & = 70.56 \\ (23 - 28.4)^2 & = 29.16 \\ (26 - 28.4)^2 & = 5.76 \\ (27 - 28.4)^2 & = 1.96 \\ (30 - 28.4)^2 & = 2.56 \\ (31 - 28.4)^2 & = 6.76 \\ (35 - 28.4)^2 & = 43.56 \\ (35 - 28.4)^2 & = 43.56 \\ \end{align*}$$ Step 3: Calculate the variance by finding the average of these squared differences. $$\text{Variance} = \frac{70.56 + 29.16 + 5.76 + 1.96 + 2.56 + 6.76 + 43.56 + 43.56}{8} = \frac{203.88}{8} = 25.49$$ Step 4: The standard deviation is the square root of the variance. $$\text{Standard Deviation} = \sqrt{25.49} \approx 5.0$$ ### Final Answers: - **Mean** = 28.4 - **Median** = 28.5 - **Mode** = 35 - **Standard Deviation** = 5.0 Would you like any further details on these calculations? ### Relative Questions: 1. How is the mean different from the median? 2. Why is the mode not always a good measure of central tendency? 3. How does the standard deviation help in understanding the data spread? 4. What happens to the mean and median if an outlier is added to the data set? 5. Can a data set have more than one mode? If yes, what is it called? 6. Why is the median sometimes preferred over the mean? 7. How do you calculate the variance from the standard deviation? 8. What effect does a larger standard deviation have on the data? **Tip:** Always check the data set for outliers before calculating the mean, as outliers can skew the results significantly.