Let's solve the problem step by step using the data provided.

The dataset is: $60, 54, 58, 58, 64, 55, 62, 59, 61, 58, 59, 61, 55, 62, 75$

Part a: Finding the quartiles

1. Step 1: Arrange the data in ascending order: $54, 55, 55, 58, 58, 58, 59, 59, 60, 61, 61, 62, 62, 64, 75$

2. Step 2: Find the quartiles.

   • First quartile (Q1): This is the median of the lower half of the data (excluding the median of the full set). For the ordered list, the lower half is: $54, 55, 55, 58, 58, 58, 59$ The median of this subset is 58, so $Q1 = 58$.

   • Second quartile (Q2): This is the median of the entire dataset. Since there are 15 values, the median is the 8th value in the ordered list, which is 59. So $Q2 = 59$.

   • Third quartile (Q3): This is the median of the upper half of the data. The upper half is: $60, 61, 61, 62, 62, 64, 75$ The median of this subset is 62, so $Q3 = 62$.

Part b: Find the interquartile range (IQR)

The interquartile range is calculated as: $IQR = Q3 - Q1 = 62 - 58 = 4$

Part c: Identify any outliers

To find outliers, use the formula: $\text{Lower bound} = Q1 - 1.5 \times IQR = 58 - 1.5 \times 4 = 52$ $\text{Upper bound} = Q3 + 1.5 \times IQR = 62 + 1.5 \times 4 = 68$ Any data point below 52 or above 68 is considered an outlier. The only value above 68 is 75, so 75 is an outlier.

Summary:

• $Q1 = 58$
• $Q2 = 59$
• $Q3 = 62$
• IQR = 4
• Outlier: 75

Would you like any further details or explanations?

Here are 5 related questions:

1. How is the median different from the mean in a dataset?
2. How can you visually represent quartiles using a box plot?
3. What is the significance of the interquartile range in data analysis?
4. How do you determine extreme outliers beyond the regular outlier threshold?
5. In what types of datasets are quartiles most useful?

Tip: Quartiles and IQR are very useful in identifying skewness or outliers in your data. Always check them before calculating the mean or making assumptions about the data's distribution.