Let's break down and solve the problem step by step.

Dataset:

$4, 6, 5, 6, 2, 7, 6, 8$

Step 1: Sort the data in ascending order:

$2, 4, 5, 6, 6, 6, 7, 8$

(a) Finding the Upper Quartile (Q3) and Lower Quartile (Q1)

To find the quartiles, we need to determine the positions of the first quartile (Q1) and the third quartile (Q3) in the dataset.

• Median (Q2): The median is the middle value of the dataset. Since there are 8 numbers (an even count), we take the average of the 4th and 5th values. $\text{Median} = \frac{6 + 6}{2} = 6$

• Lower Quartile (Q1): The lower quartile is the median of the lower half of the data. The lower half is: $2, 4, 5, 6$. The median of this subset is: $Q1 = \frac{4 + 5}{2} = 4.5$

• Upper Quartile (Q3): The upper quartile is the median of the upper half of the data. The upper half is: $6, 6, 7, 8$. The median of this subset is: $Q3 = \frac{6 + 7}{2} = 6.5$

Summary of Quartiles:

• Lower Quartile (Q1) = 4.5
• Upper Quartile (Q3) = 6.5

(b) Finding the Range and Intermediate Range

• Range: The range is the difference between the maximum and minimum values of the dataset. $\text{Range} = 8 - 2 = 6$

• Intermediate Range: The intermediate range is the difference between the upper quartile (Q3) and the lower quartile (Q1). $\text{Intermediate Range} = Q3 - Q1 = 6.5 - 4.5 = 2$

Summary of Range and Intermediate Range:

• Range = 6
• Intermediate Range = 2

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Would you like a more detailed explanation of any of these steps? Or do you have additional questions?

Here are some related questions to explore further:

1. How do you find the median of a dataset with an odd number of elements?
2. What is the significance of quartiles in statistical analysis?
3. How do outliers affect the calculation of quartiles?
4. What is the difference between the interquartile range and the range?
5. How do you interpret the meaning of the upper and lower quartiles in a dataset?

Tip: When working with quartiles, always make sure the data is sorted in ascending order before calculating Q1, Q2, and Q3. This ensures accurate results!