Let's work through the problem using the data set provided. The data set is:

59, 63, 62, 65, 66, 61, 54, 63, 55, 59, 58, 57, 56, 58, 80

Step 1: Organize the data in ascending order

The first step in finding quartiles is to sort the dataLet's work through the problem using the data set provided. The data set is:

59, 63, 62, 65, 66, 61, 54, 63, 55, 59, 58, 57, 56, 58, 80

Step 1: Organize the data in ascending order

The first step in finding quartiles is to sort the

54, 55, 56, 57, 58, 58, 59, 59, 61, 62, 63, 63, 65, 66, 80

Step 2: Find the Quartiles

• Q1 (First Quartile): This is the median of the lower half of the data.
  Lower half: 54, 55, 56, 57, 58, 58, 59
  The median of this half is 57 (middle value).

• Q2 (Second Quartile or Median): The median of the entire data set.
  Full set: 54, 55, 56, 57, 58, 58, 59, 59, 61, 62, 63, 63, 65, 66, 80
  The middle value is 59 (the 8th value in a sorted set of 15).

• Q3 (Third Quartile): This is the median of the upper half of the data.
  Upper half: 59, 61, 62, 63, 63, 65, 66, 80
  The median of this half is 63.

Step 3: Find the Interquartile Range (IQR)

The IQR is calculated by subtracting Q1 from Q3:

$IQR = Q3 - Q1 = 63 - 57 = 6$

Step 4: Identify any Outliers

Outliers are typically identified using the following rule:

• Lower Bound: $Q1 - 1.5 \times IQR$
• Upper Bound: $Q3 + 1.5 \times IQR$

Let’s calculate the bounds:

• Lower Bound: $57 - 1.5 \times 6 = 57 - 9 = 48$
• Upper Bound: $63 + 1.5 \times 6 = 63 + 9 = 72$

Any data points outside the range [48, 72] are considered outliers. The only value outside this range is 80, which is an outlier.

Summary of Results:

a. Quartiles:

• $Q1 = 57$
• $Q2 = 59$
• $Q3 = 63$

b. Interquartile Range (IQR):

• $IQR = 6$

c. Outliers:

• 80 is an outlier.

Would you like more details or have any questions? Here are 5 related questions to expand on:

1. How do you calculate percentiles for a data set?
2. How are outliers identified using different methods?
3. What other methods exist to visualize IQR and quartiles?
4. How do different data sets affect the IQR and quartile values?
5. How can skewness in data affect the calculation of quartiles?

Tip: The IQR is particularly useful in identifying outliers and understanding the spread of data. Keep it in mind when interpreting datasets with large variations.