Let's work through the problem step-by-step, starting with calculating the quartiles, the interquartile range (IQR), and then using them to find the lower and upper fences.

Given Cholesterol Levels:

$288, 234, 500, 294, 210, 236, 186, 282, 276, 280, 280, 236$

Step 1: Organize the data in ascending order

$186, 210, 234, 236, 236, 276, 280, 280, 282, 288, 294, 500$

Step 2: Calculate the Quartiles

The quartiles divide the data into four equal parts. We need to find $Q_1$, $Q_2$ (median), and $Q_3$.

• $Q_2$ (Median): This is the middle value of the dataset. Since there are 12 data points (even number), the median is the average of the 6th and 7th values: $Q_2 = \frac{276 + 280}{2} = \frac{556}{2} = 278$

• $Q_1$ (First Quartile): This is the median of the lower half of the data (excluding the median itself). The lower half is: $186, 210, 234, 236, 236, 276$ The median of this group is the average of the 3rd and 4th values: $Q_1 = \frac{234 + 236}{2} = \frac{470}{2} = 235$

• $Q_3$ (Third Quartile): This is the median of the upper half of the data (excluding the median itself). The upper half is: $280, 280, 282, 288, 294, 500$ The median of this group is the average of the 3rd and 4th values: $Q_3 = \frac{282 + 288}{2} = \frac{570}{2} = 285$

Step 3: Compute the IQR (Interquartile Range)

The IQR is the difference between $Q_3$ and $Q_1$: $\text{IQR} = Q_3 - Q_1 = 285 - 235 = 50$

Step 4: Calculate the Lower and Upper Fences

The lower and upper fences are used to detect outliers.

• Lower Fence: $\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR} = 235 - 1.5 \times 50 = 235 - 75 = 160$

• Upper Fence: $\text{Upper Fence} = Q_3 + 1.5 \times \text{IQR} = 285 + 1.5 \times 50 = 285 + 75 = 360$

Summary of Results:

• $Q_1 = 235$
• $Q_2 = 278$
• $Q_3 = 285$
• IQR = 50
• Lower Fence = 160
• Upper Fence = 360

Explanation of Errors in Original Problem:

• The quartiles seem to have been miscalculated (especially $Q_3$).
• The IQR and the fences were incorrect because they rely on the correct quartiles.

Would you like further details or have any other questions?

Here are 5 related questions to expand your knowledge:

1. What are the uses of the interquartile range in statistical analysis?
2. How do quartiles help in detecting outliers in a data set?
3. Can data points outside the fences always be considered outliers? Why or why not?
4. How does the interquartile range differ from standard deviation in measuring data spread?
5. What happens if the data set has an odd number of values when calculating quartiles?

Tip: Always sort the data first when calculating statistical measures like quartiles or percentiles to avoid errors.